a vygotskian perspective on teacher development
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PROSPECTIVE TEACHERS’ EMERGING PEDAGOGICAL CONTENT KNOWLEDGE DURING THE PROFESSIONAL
SEMESTER: A VYGOTSKIAN PERSPECTIVE ON TEACHER DEVELOPMENT
byMARIA LYNN BLANTON
A dissertation submitted to the Graduate Faculty of North Carolina State University
in partial fulfillment of the requirements for the Degree of
Doctor of Philosophy
MATHEMATICS EDUCATION
Raleigh 1998
APPROVED BY:
Dr. Glenda S. Carter Dr. Jo-Ann D. Cohen
Dr. Lee V. Stiff Dr. Karen S. Norwood Co-Chair of Advisory
Committee
Dr. Sarah B. Berenson
Co-Chair of Advisory Committee
DEDICATION
To my family.
PERSONAL BIOGRAPHY
The author was born August 7, 1967, to Tommy and Patricia
Blanton. She was raised in Willard, NC. She received her Bachelor of
Arts degree in mathematics with secondary teacher certification and
Master of Arts degree in mathematics from the University of North
Carolina-Wilmington (UNCW).
After teaching at UNCW, she moved to Raleigh, NC, to attend
graduate school at North Carolina State University. Here, she
received her Ph. D. in Mathematics Education in 1998. While a
student, she worked as a teaching assistant in the Mathematics
Department and as a research assistant in the Center for Research
in Mathematics and Science Education.
ACKNOWLEDGMENTS
I would like to thank my family for their continued support
through all my years of school. I am especially grateful to have
parents that I can count on for anything and everything. They have
always provided a weekend haven from the rigors of graduate
school. Lisa and Joey have helped maintain my perspective through
laughter. My niece, Rachel, and nephew, Joseph, have reminded me
that the most important things in life are not always measured by
academic success.
I thank Dr. Wendy Coulombe for “paving the way” for me. She
has been a valued friend and mentor. I thank Dr. Draga Vidakovic
and Dr. Susan Westbrook for being unofficial committee members.
Their advice has always been insightful and challenging.
I would like to thank members of my committee, Dr. Lee V.
Stiff, Dr. Jo-Ann Cohen, and Dr. Glenda Carter, for being a part of
this process. I extend a special thanks to Dr. Carter for our
numerous impromptu discussions on Vygotsky. She was a
tremendous “more knowing other”.
I would like to thank my co-chair, Dr. Karen Norwood, for her
unique contribution. She motivates me to pursue my own practice
with unapologetic enthusiasm. To this end, she was always willing to
extend her expertise, as well as her classroom supplies.
Most importantly, I would like to thank my major advisor, Dr.
Sally Berenson. She introduced me to a national and international
research community in mathematics education through an extensive
apprenticeship in the Center for Research in Mathematics and
Science Education. It has been an invaluable experience. Most
especially, she placed an intellectual trust in me throughout the
dissertation process. I sincerely appreciate that trust, as well as the
guidance and encouragement that accompanied it.
TABLE OF CONTENTS
Page
LIST OF
TABLES.......................................................................................................
...ix
LIST OF
FIGURES....................................................................................................
.....x
INTRODUCTION.........................................................................................
.................1
PART I: LITERATURE
REVIEW...............................................................................8
Theoretical Framework....................................................................................8
Vygotsky’s Sociocultural Theory of Learning.............................................9
General Genetic Law of Cultural Development..................................10
Psychological Tools and Signs..................................................................11
The Role of Language.................................................................................12
Social Interactions...................................................................................
....13The Zone of Proximal
Development......................................................14 Implications of Vygotsky’s Sociocultural Theory
for this Study............................................................................16
Teacher Education..........................................................................................
..17Teachers’ Beliefs and
Knowledge.............................................................17Learning How to Teach
Mathematics.....................................................20Teacher Development in
Context............................................................21Classroom
Interactions....................................................................................23
Implications.......................................................................................................26
The Nature of Qualitative Inquiry................................................................27
In-Depth Interviewing................................................................................2
8Participant
Observation..............................................................................29Teaching
Experiments................................................................................30
PART II: METHODOLOGY.........................................................................................
33
Methodological Framework...........................................................................33
Participants.........................................................................................................35
Data Collection..........................................................................................
........35Data
Analysis.....................................................................................................38
Role of the Researcher.....................................................................................4
0
PART III: MATHEMATICAL DISCOURSE IN A PROSPECTIVE TEACHER’S CLASSROOM: THE CASE OF A DEVELOPING
PRACTICE......................................................................................................................42
Abstract...............................................................................................................43
Introduction......................................................................................................44
Teacher Learning Through Classroom Discourse....................................46
Process of Inquiry.............................................................................................4
9The Research
Setting..................................................................................49Collecting the
Data......................................................................................50Analyzing Classroom
Discourse...................................................................51Pattern and Function in Teacher-Student
Talk....................................51Process of
Analysis......................................................................................54Findings and
Interpretations.........................................................................56Early Pattern and Function in Classroom
Discourse...........................56Early Pattern and Function in Resolving Students’
Mathematical Dilemmas.......................................................57Early Pattern and Function in Teaching a New
Concept...............63On Early Discourse and Mary Ann’s
Practice....................................73Indications of an Emerging Practice: Change in Pattern
and Function............................................................................75
The Problem-Solving Day.....................................................................75Moving Forward in Classroom Discourse: Learning
to Listen.....................................................................................87
Mary Ann’s Students: More Knowing Others?....................................93
Discussion..........................................................................................................95
References..........................................................................................................98Appendix..........................................................................................................102
PART IV: THE CYCLE OF MEDIATION: A TEACHER EDUCATOR’SEMERGING PEDAGOGY..........................................................................................107
Abstract.............................................................................................
................108Introduction.......................................................................................
.............109Rethinking the Role of Supervision: Education or
Evaluation?........110Collecting the Data: The Cycle of
Mediation.......................................113Pedagogy of the Teaching
Episodes.......................................................116Data
Analysis...................................................................................................118
Findings and Interpretations.......................................................................119
Instructional Conversation in Teaching Episodes with Mary
Ann.....................................................................119Activating, Using, or Providing Background Knowledge
and Relevant Schemata......................................................120
Thematic Focus for the Discussion...................................................120
Direct Teaching, as Necessary.............................................................123
Minimizing Known-Answer Questions in the Course ofthe
Discussion.......................................................................124Teacher Responsivity to Student
Contributions...........................124Connected Discourse, with Multiple and Interactive
Turns on the Same Topic...................................................127
A Challenging but Nonthreatening Environment......................129
Instructional Conversation in Retrospect: More on the
Problem-Solving Day...........................................................130
Discussion.........................................................................................................131
References........................................................................................................134
Appendix..........................................................................................................137
LIST OF
REFERENCES.............................................................................................
.141
APPENDIX..................................................................................................
..................155
LIST OF TABLES
Page
PART IV: THE CYCLE OF MEDIATION: A TEACHER
EDUCATOR’S EMERGING PEDAGOGY
1. Conversational time used by participants in the teaching
episodes.............................................................................................
...124
2. Conversational time given to subject code during teaching
episodes.............................................................................................
...129
LIST OF FIGURES
Page
PART I: LITERATURE REVIEW
1. Higher mental functioning: Vygotsky’s general
genetic law of cultural
development..........................................................11
PART II: METHODOLOGY
2. The cycle of mediation in an emerging practice of
teaching.................38
PART IV: THE CYCLE OF MEDIATION: A TEACHER EDUCATOR’S
EMERGING PEDAGOGY
1. The cycle of mediation in an emerging practice of
teaching................116
ABSTRACT
BLANTON, MARIA LYNN. Prospective Teachers’ Emerging
Pedagogical Content Knowledge During the Professional Semester:
A Vygotskian Perspective on Teacher Development. (Under the
direction of Sarah B. Berenson and Karen S. Norwood.)
This investigation adopts an interpretive approach to study a
prospective middle school mathematics teacher’s emerging
pedagogical content knowledge during the professional semester.
Vygotsky’s (1978) sociocultural perspective provides the theoretical
framework for the study. Specifically, Vygotsky’s assertion that
higher mental functioning is directly mediated through social
interactions focused this study on the intermental context in which
the prospective teacher’s practice develops during the professional
semester, or student teaching practicum.
The nature of mathematical discourse embedded in social
interactions in the prospective teacher’s classroom was analyzed as
a window into the prospective teacher’s construction of knowledge
about teaching mathematics. The role of students as more knowing
others of the classroom norms for doing mathematics and how that
mediated the teacher’s practice was considered. Analysis of pattern
and function of classroom discourse substantiated an emerging
practice, as the prospective teacher’s obligations in the classroom
transitioned from funneling students to her interpretation of a
problem to arbitrating students’ ideas.
This study also explored the pedagogy of educative supervision
and the consequent role of the university supervisor in opening the
prospective teacher’s zone of proximal development. Classroom
observations by the supervisor, teaching episode interviews between
the supervisor and the prospective teacher, and focused journal
reflections by the prospective teacher, were coordinated in a process
of supervision postulated here as the cycle of mediation.
Understanding what interactions between the university
supervisor and prospective teacher might resemble in order to
promote the prospective teacher’s development within her zone was
central to this study. The resulting pedagogy of the teaching
episodes was consistent with instructional conversation (Gallimore &
Goldenberg, 1992). In this case, instructional conversation seemed
to open the prospective teacher’s zone so that her understanding of
teaching mathematics could be mediated with the assistance of a
more knowing other. This, together with the cycle of mediation,
suggests an alternative model for helping teachers develop their
craft in the context of practice.
INTRODUCTION
Historically, mathematics education has entertained diverse
views in an almost eclectic move toward a unified theory of learning.
Indeed, advances in cognitive psychology have prompted a shift from
stimulus-response models in which learning is defined by students’
perfunctory reactions to stimuli, to meaning-based models such as
constructivism, in which students are seen as actively and
individually creating their own knowledge (Noddings, 1990; von
Glasersfeld, 1987). Recently, as disciplines such as anthropology and
sociology have joined the quest for a comprehensive theory of
learning, emphasis on the more prevalent Western tradition of
individual knowledge construction has broadened to include the role
of culture and context in this process as well (e. g., Cobb &
Bauersfeld, 1995; Eisenhart & Borko, 1991; Ernest, 1994; Saxe,
1992; Shulman, 1992). The resultant theory, generally described as
social constructivism, has become a watchword for those who
espouse constructivist views that recognize contributions from social
processes and individual sense making in learning (Ernest, 1994).
For the most part, Vygotskian and Piagetian theories of mind have
dominated thinking in this area as scholars debate the primacy of
the social versus the individual in knowledge construction (e. g.,
Cole & Wertsch, 1994; Confrey, 1995; Ernest, 1995; Shotter, 1995).
In some cases, such debates have been discarded in favor of
theoretical perspectives that coordinate social and individual
domains in a complementary fashion (Cobb, Yackel, & Wood, 1993).
Mathematics education has led reform efforts in its attempts to
incorporate recent research in such disciplines as cognitive
psychology into an existing knowledge base to produce a codified
body of principles, or standards, for teaching and learning
mathematics. Most notably, the National Council of Teachers of
Mathematics (NCTM) Curriculum and Evaluation Standards for
School Mathematics (1989), which has theoretical roots in
constructivism, is grounded in two decades of research on students’
thinking about mathematics (Simon, 1997). According to Simon, a
strong research base on teacher development that parallels national
reform efforts in students’ mathematical development is currently
needed in the mathematics education community. It is not enough to
understand the process of learning mathematics; mathematics
educators must also understand the process of teaching mathematics
in reform-minded ways. Thus, the question becomes how can teacher
education programs integrate research in such disciplines as
cognitive psychology, sociology, and anthropology with that of
mathematics education to prepare a professional cadre of
mathematics teachers? More specifically, how can such programs
prepare inservice and prospective teachers to teach mathematics in
a manner consistent with the recommendations of the NCTM
Curriculum and Evaluation Standards? The NCTM Professional
Standards for Teaching Mathematics (1990) offers a timely response
to this question. Its stated purpose is to provide a set of standards
that
promotes a vision of mathematics teaching, evaluating
mathematics teaching, the professional development of
mathematics teachers, and responsibilities for professional
development and support, all of which would contribute to the
improvement of mathematics education as envisioned in the
Curriculum and Evaluation Standards (p. vii).
Furthermore, it advocates five major shifts in classroom perspectives
in order to promote students’ intellectual autonomy. In particular,
teachers’ thinking needs to shift
(a) toward classrooms as mathematical communities-away
from classrooms as simply a collection of individuals;
(b) toward logic and mathematical evidence as verification-
away from the teacher as the sole authority for right answers;
(c) toward mathematical reasoning-away from merely
memorizing procedures;
(d) toward conjecturing, inventing, and problem-solving-away
from an emphasis on mechanistic answer-finding;
(e) toward connecting mathematics, its ideas, and its
applications-away from treating mathematics as a body of
isolated concepts and procedures (p. 3).
Such recommendations reflect critical insights into teaching
mathematics and are consistent with the Curriculum and Evaluation
Standards.
Various long-term research agendas in mathematics education
directed towards prospective and inservice teachers are working to
address the need for a reform-driven research base in teacher
development (e. g., Ball, 1988; Berenson, Van der Valk, Oldham,
Runesson, Moreira, & Broekman, 1997; Carpenter, Fennema,
Peterson, & Carey, 1988; Cobb, Yackel, & Wood, 1991; Eisenhart,
Borko, Underhill, Brown, Jones, & Agard, 1993; Feiman-Nemser,
1983; National Center for Research on Teacher Education, 1988;
Schram, Wilcox, Lappan, & Lanier, 1989; Shulman, 1986; Simon,
1997). One such program has identified seven domains that
constitute teachers’ professional knowledge as content knowledge,
pedagogical content knowledge, general pedagogical knowledge,
knowledge of educational contexts, knowledge of curriculum,
knowledge of learners, and knowledge of educational aims
(Shulman, 1987). Shulman’s model continues to provide a conceptual
framework for other studies on teaching. Indeed, a number of
researchers in mathematics education (e. g., Ball, 1990; Berenson, et
al., 1997; Borko, Eisenhart, Brown, Underhill, Jones, & Agard, 1992;
Even & Tirosh, 1995; McDiarmid, Ball, & Anderson, 1989) recognize
that understanding of these knowledge domains, as well as the
consequent role of teacher education programs in teacher
preparation, is currently underdeveloped. They have accepted the
challenge this offers by studying various strands within each domain
as well as the connections that exist among them.
Of the seven components of this knowledge base for teaching,
pedagogical content knowledge was the focus of this study. Shulman
(1987) defines such knowledge as
that special amalgam of content and pedagogy that is uniquely
the province of teachers.... [It is] the blending of content and
pedagogy into an understanding of how particular topics,
problems, or issues are organized, represented, and adapted to
the diverse interests and abilities of learners, and presented
for instruction (p. 8).
Pedagogical content knowledge is recognized among mathematics
educators as playing a central role in one’s development from
learning mathematics to teaching mathematics (Ball, 1990; Borko, et
al., 1992; Even, 1993). Additionally, they acknowledge that our
understanding of this domain, as well as the integrated manner in
which it exists in the teaching process, is incomplete. Based on the
premise that the professional semester, or student teaching
practicum, is a pivotal context in which prospective teachers begin
to construct pedagogical content knowledge, this study considered
how the prospective mathematics teacher’s practice emerges during
this stage.
In order to understand the construction of pedagogical content
knowledge, I appealed to the theoretical lens of social
constructivism. Viewing mind metaphorically as social and
conversational, Ernest (1994) posits that people are “formed through
their interactions with each other (as well as by their internal
processes) in social contexts” (p. 69). This is no less true for
prospective teachers during the student teaching practicum. Indeed,
Vygotsky’s (1986) assertion that higher mental functions are directly
mediated through social interactions suggests that the prospective
teacher’s transition from mathematics student to mathematics
teacher does not occur apart from human interaction; rather, as a
result of it.
Such transitions can be characterized as a process of
acculturation resulting from one’s (i. e., the prospective teacher’s)
development within the zone of proximal development. The zone of
proximal development is defined by Vygotsky (1978) as “the distance
between the [individual’s] actual developmental level as determined
through independent problem solving and the level of potential
development as determined through problem solving under adult
guidance or in collaboration with more capable peers” (p. 86). This
suggests the importance of instructional assistance in the
prospective teacher’s development.
This study is an investigation of the prospective middle school
mathematics teacher’s emerging practice of teaching during the
professional semester. In particular, I first considered the nature of
mathematical discourse, or conversation, embedded in social
interactions in the prospective teacher’s mathematics classroom as
preliminary to the broader context of teacher development. The
nature of such discourse was expected to provide a window into the
prospective teacher’s construction of knowledge about teaching
mathematics. Also, I examined the university supervisor’s role as a
more knowing other in the prospective teacher’s emerging practice.
Specifically, I considered what the pedagogy of supervision might
resemble in order to open the prospective teacher’s zone of proximal
development and effect a change in practice. Thus, the following
questions were formulated to guide this research:
1. What is the nature of mathematical discourse in the
prospective teacher’s mathematics classroom during the
professional semester?
2. How does the university supervisor influence the
prospective teacher’s emerging practice of teaching
through the zone of proximal development?
LITERATURE REVIEW
This chapter begins with a discussion of the social construction
of knowledge as a theory of learning. It includes a detailed
examination of the sociocultural theory of Lev Vygotsky, which
provided the theoretical framework for this study. Attention is given
to the basic tenets of Vygotsky’s theory as well as various constructs
associated with it. Linkages between his theory and this study are
established. A review of current literature on the preparation and
development of teachers follows this. In connection with this, the
role of classroom interactions in the social construction of
knowledge is examined. Implications of this study in addressing the
limitations of existing research in teacher education are discussed.
Finally, the process of qualitative inquiry is described to support this
choice of research paradigm for the study.
Theoretical Framework
Shulman (1992) wrote that “knowledge is socially constructed
because it is always emerging anew from the dialogues and
disagreements of its inventors” (p. 27). This suggests an inherent
complexity of social constructivism. That is, social constructivism is
difficult to precisely define because it is subject to the varied
experiences and biases of its inventors. Ernest (1994) comments that
there is a “lack of consensus about what is meant by the term, and
what its underpinning theoretical bases are” (p. 63). He recognizes
that both social processes and individual sense making are central to
a social constructivist theory, and that the emphasis given to either
domain will vary depending on one’s theoretical assumptions
concerning the nature of mind. In particular, the social
constructivist’s view of mind will often have Piagetian or Vygotskian
roots, although one may rely on other perspectives more or less
compatible with these traditions. A Piagetian view prioritizes the
individual act of knowledge construction by interpreting social
processes as either secondary, or separate, but equal. Ernest
maintains that a Vygotskian theory of mind “views individual
subjects and the realm of the social as indissolubly interconnected”
(p. 69). He further explains that
mind is viewed as social and conversational because....first of
all, individual thinking of any complexity originates with, and is
formed by, internalized conversation; second, all subsequent
individual thinking is structured and natured by this origin;
and third, some mental functioning is collective (p. 69).
In this study, I have assumed a Vygotskian theory of mind. As such,
the remainder of this section will be used to outline the basic tenets
of such a theory and how it serves as the framework for this study.
Vygotsky’s Sociocultural Theory of Learning
According to Wertsch (1988), Vygotsky’s theory of mind
consists of three major themes. First, Vygotsky maintained that any
component of mental functioning is understood only by
understanding its origin and history. As Luria, a protégé of Vygotsky,
summarized,
in order to explain the highly complex forms of human
consciousness one must go beyond the human organism. One
must seek the origins of conscious activity....in the external
processes of social life, in the social and historical forms of
human existence (1981, as cited in Wertsch & Tulviste, 1996,
p. 54).
To this end, Vygotsky considered the life span development of the
individual (ontogenesis) and the development of species
(phylogenesis), as well as the associated sociocultural history. This
emphasis represented a shift from the traditional focus of his
contemporaries on the individuality of child development.
General Genetic Law of Cultural Development
Another major theme of Vygotsky’s theory is found in his
general genetic law of cultural development. This theorization of the
relationship between social and individual domains in higher mental
functioning emphasizes Vygotsky’s belief in the social formation of
mind: “Social relations or relations among people genetically
underlie all higher functions and their relationships” (Vygotsky,
1981b, as cited in Wertsch & Tulviste, 1996, p. 55). The general
genetic law of cultural development posits that an individual’s higher
(i. e., uniquely human) mental functioning originates in the social
realm, or between people, on an intermental plane. Internalization of
higher mental functions is then a process of genetic (i. e.,
developmental) transformation of lower mental functions to the
intramental plane, within the individual (Wertsch, 1988; Wertsch &
Toma, 1995). This process is illustrated in Figure 1. According to
Holzman (1996), the exact nature of this genetic transformation has
been a subject for much research. In particular, research in Soviet
psychology has produced a method of investigation known as the
microgenetic approach (from microgenesis). This approach involves
charting the transition from the intermental plane to the intramental
plane over the course of a brief social interaction in order to study
the process of change that occurs.
Figure 1. Higher mental functioning: Vygotsky’s general genetic law
of cultural development.
Psychological Tools and Signs
Finally, Vygotsky believed that higher mental functioning is
mediated by socioculturally-evolved tools and signs (Wertsch, 1988).
In particular, Vygotsky (1986) addressed human use of technical, or
physical, tools to illustrate the role of psychological tools in higher
mental functioning. He maintained that a physical tool acts as a
mediator between the human hand and the object on which it acts in
order to control natural, or environmental, processes. In an
analogous manner, psychological tools such as gestures, language
systems, mnemonic devices, and algebraic symbol systems, serve to
control human behavior and cognition by “transforming the natural
human abilities and skills into higher mental functions” (p. xxv).
According to Vygotsky, “humans master themselves from the outside
- through psychological tools” (p. xxvi).
Vygotsky studied signs as a special form of psychological tools
(Minick, 1996). Wertsch and Toma (1995) recognize this form as
well: “Of particular interest to [Vygotsky] were signs, which
constituted a broad category of mediational means used to organize
one’s own or others’ actions” (p. 163). These artificial stimuli include
such symbolic formations as social languages, mathematical systems,
and diagrams. Bakhurst (1996) describes tying a knot in a
handkerchief as a sign to invoke later rememberings. In this simple
illustration, the knot serves as a sign to control one’s behavior.
The Role of Language
Vygotsky (1986) viewed language as the most powerful
psychological tool for mediating higher mental functions. It is the
primary medium through which thought develops, making possible
the transition from the intermental plane to the intramental plane.
Furthermore, as a higher mental function, language is also subject to
the mediating effect of tools. Concerning this duality, Holzman
(1996) explains that the
dialectical role of speech is that it plays a part in defining the
task setting; this activity redefines the situation, and in turn,
speech is redefined. Language is viewed as both tool and result
of interpersonal [i. e., intermental] and intrapersonal [i. e.,
intramental] psychological functioning (p. 91).
In other words, language is unique in that it is both a mediating tool
and a mediated function.
Social Interactions
Vygotsky’s belief in the social origins of higher mental
functions and the mediating role of language in their development
underscores the importance of social interactions. Indeed, Vygotsky
argued that social interactions are the basis for an individual’s
development (Holzman, 1996). Minick (1996) explains that
Vygotsky turned to the primary function of speech as a means
of communication. [He] argued that the higher voluntary forms
of human behavior have their roots in social interaction, in the
individual’s participation in social behaviors that are mediated
by speech. It is in social interaction, in behavior that is being
carried out by more than one individual, that signs first
function as psychological tools in behavior. The individual
participates in social activity mediated by speech, by
psychological tools that others use to influence his behavior
and that he uses to influence the behavior of others (p. 33).
As an illustration, consider teaching a child to add fractions.
In the process of instruction, the teacher uses tools (e. g., language,
figural diagrams, and the real number system) to mediate the child’s
behavior or thinking. Once the child has appropriated this skill, he or
she then uses it in his or her own mathematical activity and
sometimes to influence the activity of peers. In this scenario, the
child’s development occurs within the context of social interactions.
While this illustration implies human-human interaction as a defining
characteristic of social interactions, participants in social
interactions are interpreted more broadly here to include
representations of ideas, such as those embodied in reading
materials. In this situation, the reader’s thinking is mediated through
written speech. Wilson, Teslow, and Taylor (1993) address this,
suggesting that the interactions between teacher and student can be
extended to “include interactions between learners and technology-
based tools and agents” (p. 81).
The Zone of Proximal Development
The zone of proximal development is one of the central
propositions of Vygotsky’s sociocultural theory. Daniels (1996)
describes this theoretical construct as the setting in which the social
and individual domains meet. Wertsch and Tulviste (1996) further
explain that the zone of proximal development has “powerful
implications for how one can change intermental, and hence
intramental, functioning” (p. 57). Change results from tool-mediated
activity such as instruction, that is, assistance by a more knowing
other offered through social interactions with the student. In turn,
instruction creates the zone of proximal development, which
stimulates inner developmental processes (Hedegaard, 1996). The
teacher’s task is to provide meaningful instructional experiences that
enable the student to bridge his or her zone of proximal
development. As such, the zone of proximal development is unique in
that it “connects a general psychological perspective
on...development with a pedagogical perspective on instruction” (p.
171).
A stringent interpretation of Vygotsky’s definition of the zone
of proximal development requires an adult or more capable peer to
foster one’s development. However, Oerter (1992) distinguishes
three contexts which can create one’s zone of proximal development:
intentional instruction (such as that given by a teacher or parent),
stimulating environments (such as books or materials for painting),
and play. He cites Vygotsky’s observations that children at play
create their own zones of proximal development: “In play the child
tries as if to accomplish a jump above the level of his ordinary
behavior” (Vygotsky, 1966, as cited in Oerter, 1992, p. 188). The
common thread is the presence of help in one’s construction of
knowledge. According to Taylor (1993), Vygotsky also suggested that
a student’s interactions with materials (e. g., manipulatives) can
enable that student to bridge the zone of proximal development for
deeper understanding. One can speculate that, had Vygotsky lived
long enough, his definition may have reflected this.
Implications of Vygotsky’s Sociocultural Theory for this Study
Eisenhart (1991) describes a theoretical framework as a
“structure that guides research by relying on a formal theory; that is,
the framework is constructed by using an established, coherent
explanation of certain phenomena and relationships” (p. 205). In this
sense, Vygotsky’s sociocultural theory guided my investigation of the
prospective teacher’s emerging practice. As a formal theory, it
provided an established language for communicating research, as
well as an accepted format for investigation. More specifically,
Vygotsky’s general genetic law of cultural development directed me
to social interactions as a forum for the prospective teacher’s
construction of pedagogical content knowledge. Furthermore, his
emphasis on the mediating affect of tools and signs, particularly
language, led me to investigate the role of language in that process.
Finally, Vygotsky’s construct of the zone of proximal development
supports the use of intentional instruction during supervision to
influence the prospective teacher’s development. According to
Manning and Payne (1993), “The mechanism for growth in the zone
is the actual verbal interaction with a more experienced member of
society. In the teacher education context, this more experienced
person is likely to be a supervising teacher, college supervisor,
teacher educator, or a peer who is at a more advanced level in the
teacher education program” (as cited in Jones, Rua, & Carter, 1997,
p. 6).
Teacher Education
As new theories of learning emerge, it becomes necessary to
rethink how we prepare prospective and inservice teachers. The
purpose of this section is to acquaint the reader with current studies
in teacher education with this objective. Cooney (1994) reports that
research in teacher education, more and more frequently situated in
interpretivist frameworks, emphasizes teachers’ cognitions and the
factors influencing those cognitions. He includes research on
teachers’ beliefs and conceptions, teachers’ knowledge of
mathematics, and learning how to teach, in this emphasis.
Additionally, Cooney credits the preeminence of constructivism as an
epistemological foundation of mathematics education for efforts to
reform teaching and teacher education. Regarding such reform,
Simon (1997) addresses the need for models of teaching consistent
with constructivist perspectives to serve as research frameworks for
mathematics teacher development. He postulates the Mathematics
Teaching Cycle, which characterizes the “relationships among
teachers’ knowledge, goals for students, anticipation of student
learning, planning, and interaction with students” (p. 76), as one
such framework. According to Cooney, Simon’s purpose is to
articulate explicit teaching principles based on constructivism “with
the intent that these principles will serve as organizing agents for
both research and development activities in teacher education” (p.
613).
Teachers’ Beliefs and Knowledge
Shulman’s knowledge base for teaching, developed through
research on how prospective teachers “learn to transform their own
understanding of subject matter into representations and forms of
presentation that make sense to students” (Shulman & Grossman,
1988, as cited in Brown & Borko, 1992, p. 217), has often provided a
framework for studying teacher development. Within this knowledge
base, content knowledge and pedagogical content knowledge have
received the most attention in educational research (Brown & Borko,
1992). In particular, Even (1993) has studied prospective secondary
mathematics teachers’ subject matter knowledge of the function
concept and its relationship to their pedagogical content knowledge.
A conclusion was that prospective teachers have a limited
understanding of functions, which is evidenced in their instructional
decisions. In addition, Even and Tirosh (1995) have investigated the
interconnections between secondary mathematics teachers’ subject
matter knowledge and knowledge about students and teachers’ ways
of presenting the subject matter. Their interviews with participants
revealed the need to raise the sensitivity of teachers to students’
thinking about mathematics. They further concluded that teacher
education programs should incorporate specific concepts from the
school curriculum to ensure that prospective teachers’ subject
matter knowledge is “sufficiently comprehensive and articulated for
teaching” (p. 18).
The National Center for Research on Teacher Education
(NCRTE) has implemented various research programs focusing on
elementary teacher preparation. Ball (1988) describes one project of
the NCRTE to investigate changes in prospective and inservice
teachers’ knowledge. This longitudinal study examined what
teachers are taught and what they learn, with an emphasis on
“whether and how their ideas or practices change and what factors
seem to play a role in any such changes” (p. 18). To do this, they
specified four domains of a knowledge base reflective of those
identified by Shulman: subject matter knowledge, knowledge of
learners, knowledge of teaching and learning, and knowledge of
context. Of these domains, Ball has focused on elementary and
secondary mathematics teachers’ subject matter knowledge,
identifying it as a central requisite for teacher preparation (Brown &
Borko, 1992). Observing such teachers’ representations of division
at the beginning of the teacher education program, she concluded
that their subject matter knowledge was often fragmented and rule-
dependent (Ball, 1990). Furthermore, Ball and Mosenthal (1990)
found that teacher educators often place less emphasis on this
knowledge domain, thus contributing to the dilemma.
Another program of the NCRTE addressed the nature of
elementary prospective teachers’ beliefs and knowledge about
mathematics, learning mathematics, and teaching mathematics, as
well as changes that resulted from their participation in a
coordinated sequence of innovative mathematics courses and
mathematics methods courses (Schram, et al., 1989). Analyses of
this longitudinal study showed that prospective teachers’ beliefs and
knowledge about mathematics, mathematics learning, and
mathematics teaching were positively affected by the course
sequence. However, the student teaching practicum revealed a
tension between their views as adult students of mathematics and
their instructional practices with children (Brown & Borko, 1992).
Learning How to Teach Mathematics
Feiman-Nemser (1983) has examined prospective elementary
teachers’ transition to pedagogical thinking. Such a transition is
characterized by a shift in the teacher’s thinking away from the
teacher and the content and toward students’ needs. Feiman-
Nemser and colleagues concluded that, alone, prospective teachers
can rarely see beyond what they want or need to do, or what
the setting requires. They cannot be expected to analyze the
knowledge and beliefs they draw upon in making instructional
decisions, or their reasons for these decisions, while trying to
cope with the demands of the classroom” (Brown & Borko,
1992, p. 217).
They maintained that the prospective teacher’s support personnel
should be actively guiding the prospective teacher and encouraging
him or her to analyze and discuss instructional decisions. This
conclusion has powerful implications for the role of the university
supervisor as the prospective teacher’s more knowing other during
the professional semester.
Elementary and secondary prospective teachers were the focus
of a program of study by Borko and colleagues that investigated
teachers’ thinking during the planning and instructional phases of
teaching (Brown & Borko, 1992). From this study, the researchers
identified several areas affecting success in learning to teach. In
particular, successful teachers exhibited careful planning that
anticipated students’ problems and provided strategies for
overcoming them, they demonstrated strong preparation in content,
and they held the view, supported by colleagues and administrators,
that the prospective teacher is responsible for classroom events.
In a related study, Eisenhart and colleagues (1993) studied
prospective teachers’ procedural and conceptual knowledge in the
process of learning to teach mathematics for understanding. Their
investigation of one student teacher’s ideas and practices
concerning teaching for procedural and conceptual knowledge
revealed a tension between the teacher’s stated commitment and the
reality of instruction, with instruction focusing on procedural
knowledge. Such a tension was echoed by the stated beliefs and
actions of the student teacher’s support personnel. The researchers
concluded that teaching for conceptual knowledge should enjoy
consistent support from all of the professional participants in the
student teacher’s experience in order to resolve these tensions.
Teacher Development in Context
Included in this review of research on teacher preparation and
development is a research program for inservice teachers known as
the Second-Grade Classroom Teaching Project (Cobb, et al., 1991).
This study is of particular interest because of its emphasis on
knowledge construction in the context of classroom interactions.
Additionally, the researchers’ use of a classroom teaching
experiment to effect changes in teaching practices supports the use
of such methodology in this study. Embedded within a theoretical
framework of constructivism that equally emphasizes the social
negotiation of classroom norms, the Second-Grade Classroom
Teaching Project addresses second-grade students’ construction of
mathematical knowledge, as well as the development of a
constructivist-based curriculum and the preparation of elementary
teachers to teach in a manner consistent with such a curriculum.
Concerning teacher development, Cobb and colleagues speculate
that
the phenomena of implicit routines and dilemmas suggest that
teachers should be helped to develop their pedagogical
knowledge and beliefs in the context of their classroom
practice. It is as teachers interact with their students in
concrete situations that they encounter problems that call for
reflection and deliberation. These are the occasions where
teachers can learn from experience. Discussions of these
concrete cases with an observer who suggests an alternative
way to frame the situation or simply calls into question some of
the teacher’s underlying assumptions can guide the teacher’s
learning (p. 90).
They also recognize that models of teachers’ constructions of
pedagogical content knowledge are needed. Furthermore, from
looking within the classroom to determine models of children’s
constructions of mathematical knowledge, they suggest that the
appropriate setting in which to ascertain teachers’ models is also the
classroom. Their investigation of one teacher’s learning that
occurred in the mathematics classroom indicated that the teacher’s
beliefs about the nature of mathematics and learning were affected
as she resolved conflicts between her existing teaching practices and
the project’s emphasis on teaching practices that promoted students’
constructions of mathematical knowledge.
Classroom Interactions
Given the recent attention to social constructivism as an
epistemological orientation, it follows that social interactions should
be represented in the research literature. In education, the idea of
social interactions in the classroom is intrinsically bound to such an
orientation. The purpose of this section is to inform the reader of
studies on classroom interactions, as well as discussions in the
literature concerning relevant theoretical perspectives.
Bartolini-Bussi’s (1994) theoretical predilections are more
Vygotskian than Piagetian; however, she argues for the acceptance
of complementarity as the basis for theoretical and empirical
research on classroom interaction in teaching and learning.
Complementarity separates the social and individual domains, yet
attaches equal importance to both. Bartolini-Bussi advocates the
freedom to “refer to approaches that are theoretically incompatible”
rather than yield allegiance to one system (p. 128). The latter can
potentially blind the researcher to “relevant aspects of reality...or
[introduce] into the system such complications as to make it no
longer manageable” (p. 130). Others echo this approach in their own
research (e. g., Cobb & Bauersfeld, 1995; Cobb, Wood, Yackel, &
McNeal, 1992).
In her theoretical discussion of research on classroom
interactions, Bartolini-Bussi (1994) cites studies on such interactions
in mathematics teaching and learning. This includes her own
research on the relationship between social interactions and
knowledge in the mathematics classroom, based on the
Mathematical Discussion in Primary School Project (see Bartolini-
Bussi, 1991). Also mentioned is work by Balacheff (1990) that
considers social interactions to understand how students treat
refutation in the problem of mathematical proof.
Elsewhere, using a teaching experiment to investigate
children’s constructions of mathematics, Steffe and Tzur (1994)
analyzed social interactions attendant with children’s work on
fractions using computer microworlds. They extended social
interactions to mathematical interactions, with the latter including
enactment or potential enactment of children’s operative
mathematical schemes. Furthermore, they examined both nonverbal
and verbal forms of communication as constituting mathematical
interactions. Consistent with their Piagetian roots, Steffe and Tzur
concluded that social interactions contribute to children’s
mathematical constructions, but are not their primary source.
Much of the research on classroom interactions using an
interactionist perspective comes from the individual and collective
efforts of Cobb, Bauersfeld, and their colleagues (e. g., Bauersfeld,
1994; Cobb, 1995; Cobb & Bauersfeld, 1995; Cobb, et al., 1992;
Voigt, 1995). Bauersfeld (1994) characterizes the interactionist
perspective as the link between the two extremes of individualism
and collectivism. The research traditions of symbolic interactionism
and ethnomethodology are prototypical of this perspective, which
establishes teachers and students as interactively constituting the
culture of the mathematics classroom. This perspective is
distinguished from the collectivist (e. g., Vygotskian) perspective, in
which learning is a process of enculturation into an existing culture,
and the individualistic (e. g., Piagetian) perspective, in which
learning is a process of individual change. Their work, like that of
many others discussed here, is positioned within elementary school
mathematics.
In an interactional analysis of classroom mathematics
traditions, Cobb and colleagues (1992) considered what it means to
teach and learn elementary school mathematics. Their approach
assumed that “qualitative differences in...classroom mathematics
traditions can be brought to the fore by analyzing teachers’ and
students’ mathematical explanations and justifications during
classroom discourse” (p. 574). I have made a similar assumption in
the present study. That is, classroom discourse is a catalyst for
elucidating qualitative differences in the emerging classroom
traditions of prospective mathematics teachers.
Research on interactions in the mathematics classroom
suggests an interesting analogy for research in the “teaching
mathematics” classroom (Cobb, et al., 1991). Just as research on
mathematics classroom interactions offers insights into children’s
constructions of mathematical knowledge (Cobb, 1995; Steffe &
Tzur, 1994), it is theoretically feasible that interactions in the
prospective teacher’s “classroom” should provide understanding of
how knowledge about teaching mathematics is constructed. In this
context, I interpret the prospective teacher’s classroom as the
various forums during the professional semester in which his or her
pedagogical content knowledge is mediated.
Implications
As the literature suggests, there is a growing research base
concerning the development of prospective teachers, as well as the
social construction of knowledge. However, more work integrating
these two areas is needed. In mathematics education, the balance of
research on prospective teacher development rests within the
elementary teacher population. Additionally, research on the social
construction of knowledge has been dominated by children’s
constructions of mathematical knowledge. As such, social
constructivism as an interpretive framework offers a rich basis for
research in mathematics teacher education. Specifically, we need to
consider how prospective teachers of all levels of mathematics
construct their knowledge of teaching. In addition, we need to find
new ways to “guide and support teachers as they learn in the setting
of their classroom” (Wood, Cobb, & Yackel, 1991, p. 611). By
adopting a Vygotskian perspective to investigate the prospective
middle school mathematics teacher’s emerging practice during the
professional semester and how that process can be encouraged
through external support, this study has addressed some of the
limitations of the existing research.
The Nature of Qualitative Inquiry
In addressing the possibility of alternative research models
with which to study teaching, Shulman (1992) looks beyond the
traditional focus of social science research in favor of a “move
toward a more local, case-based, narrative field of study” (p. 26).
This perspective reflects a growing genre of educational research for
which qualitative inquiry is appropriate. According to Cooney (1994),
the current emphasis in education on cognition and context has
produced a “rather dramatic shift away from the use of quantitative
methodologies based on a positivist framework to that of interpretive
research methodologies” (p. 613).
Qualitative research seeks to descriptively portray some
phenomenon under investigation through a “bottom up” approach in
which an explanation of the phenomenon emerges from the data.
Sometimes referred to as grounded theory, this approach is
succinctly illustrated by Bogdan and Biklen (1992) as the piecing
together of a puzzle whose picture is not known in advance, but
rather is constructed as the researcher gathers and analyzes the
parts. To accomplish this, the qualitative researcher is uniquely
positioned within the very process of the research, a role which
necessitates that any observations be filtered through the
researcher’s own interpretive lens. Understanding involves the
assumption that the world of inquiry is a complex system in which
every detail could further explain the reality under investigation.
Typically in qualitative research, an explanation for some type
of behavior is sought through an inductive process of spontaneous,
unstructured data collection (Bogdan & Biklen, 1992). A variety of
methods are available to the researcher for this purpose, any of
which may generate copious data that must be coded and analyzed
for presentation in a manageable form. The most prevalent of these
methods are in-depth interviewing and participant observation,
supplemented at times by artifact reviews. Although used less
frequently, teaching experiments offer a unique contribution to
qualitative research methodology as well.
In-Depth Interviewing
In-depth, open-ended interviewing is an essential tool of
qualitative research in which the researcher is “bent on
understanding, in considerable detail, how people such as teachers,
principals, and students think and how they came to develop the
perspectives they hold” (Bogdan & Biklen, 1992, p. 2). It is the
foremost medium through which the researcher gains access to
events in one’s mind that are not directly observable.
Patton (1990) has suggested three approaches to structuring
an interview for research purposes: the informal conversational
interview, the general interview guide, and the standardized open-
ended interview. The informal conversational interview has the
advantage of occurring as a natural extension of ongoing fieldwork
to the extent that the participant may not perceive the interaction as
an interview. The direction of the interview depends on events
occurring in a given setting and as such, predetermined questions
are not considered. The general interview guide offers a semi-
structured approach to interviewing through a checklist of relevant
topics to be discussed in some manner with each of the participants.
The most structured of the three approaches, the standardized open-
ended interview flows from a precisely worded set of questions
posed to each of the participants for the purpose of minimizing any
variations across interviews.
In common to all three approaches is the adherence to open-
endedness. It is essential that respondents be allowed to express
their perceptions in their own words, without consulting a
preconceived set of responses and without being guided by the
wording of an interview question.
Participant Observation
Bogdan and Biklen (1992) describe participant observation as
when the researcher “enters the world of the people he or she plans
to study, gets to know, be known, and trusted by them, and
systematically keeps a detailed written record of what is heard and
observed” (p. 2). The level of the researcher’s participation will vary
depending on the goals of the study, as well as any inherent
constraints of the research site. Concerning this participatory role,
Smith (1987) suggests that the “researcher must personally become
situated in the subject’s natural setting and study, firsthand and over
a prolonged time, the object of interest” (p. 175).
Observations made in the research setting are documented
through field notes, as well as audio recordings, audiovisual
recordings, or both. Although field notes can be broadly interpreted
to mean any data collected in the process of a particular study,
Bogdan and Biklen (1992) define it more narrowly as “the written
account of what the researcher hears, sees, experiences, and thinks
in the course of collecting and reflecting on the data in a qualitative
study” (p. 107). Typically, field notes taken during an observation are
hurried accounts of the events, people, objects, activities, and
conversations that are part of the setting. Ideally, this abbreviated
version is extended immediately after an observation into a full
description that includes the researcher’s reflections about emerging
patterns and strategies for further observations. This information is
often triangulated by the collection of documents or artifacts that
are relevant to the study. These items may be personal writings,
memos, portfolios, records, articles, or photographs. The review of
such artifacts is often regarded metaphorically as an interview.
Teaching Experiments
For some mathematics educators (e. g., Ball, 1993; Cobb &
Steffe, 1983; Lampert, 1992; Thompson & Thompson, 1994), a
particular phenomenon is best understood when the participatory
role of the observer enlarges to that of teacher, evoking a classroom-
based research model in which one studies mathematics learning by
becoming the mathematics teacher. Such action research describes
a “type of applied research in which the researcher is actively
involved in the cause for which the research is conducted” (Bogdan
& Biklen, 1992, p. 223). When the active involvement alludes to the
researcher as teacher, it generally refers to a teaching experiment.
In particular, Romberg (1992) defines the teaching experiment as a
method in which “hypotheses are first formed concerning the
learning process, a teaching strategy is developed that involves
systematic intervention and stimulation of the student’s learning,
and both the effectiveness of the teaching strategy and the reasons
for its effectiveness are determined” (p. 57).
Steffe (1991) describes the teaching experiment as directed
towards understanding the progress one makes over an extended
period of time. “The basic and unrelenting goal of a teaching
experiment is for the researcher to learn the mathematical
knowledge of the involved children and how they construct it” (p.
178). While his characterization refers specifically to children
constructing mathematical knowledge, it is appropriate to extend
this notion to include other learning situations, such as prospective
teachers constructing pedagogical content knowledge.
Steffe (1983) outlines three major components of the teaching
experiment as a methodology for constructivist research: modeling,
teaching episodes, and individual interviews. He uses models to
connote an explanation formulated by the researcher to describe
how students construct mental objects. His interpretation of
Vygotsky’s methodology prioritizes the development of such models
as a goal of teaching experiments. The teaching episodes involve a
teacher, student, and witness of the teacher-student interaction. The
teacher’s role is to challenge the model, or explanation, of the
student’s knowledge and examine how that model changes through
purposeful intervention. This component is consistent with the
Vygotskian (1986) notion of creating a student’s zone of proximal
development and offering instructional assistance in order to effect
the student’s conceptual change. Finally, Steffe suggests that
teaching episodes should be followed by individual interviews, which
differ from the former only in the absence of purposeful intervention
by the teacher with the student.
Vygotsky’s (1986) studies of conceptual development in
children indicate that teaching within the context of an investigation
is not a new approach. His view that one’s intellectual ability is more
accurately described as what can be accomplished with the help of a
more knowing other than what can be accomplished when working
alone shaped the nature of his investigations, often casting him in
the role of teacher. Although “the methodology of the teaching
experiment does not apply exclusively to a particular theory”
(Skemp, 1979, as cited in Steffe, 1983, p. 470), it describes the
nature of Vygotsky’s inquiry. As such, the teaching experiment is
particularly appropriate for studies that assume a Vygotskian
theoretical framework for the purpose of understanding one’s
development.
Finally, it should be emphasized that qualitative research
requires a philosophical perspective that is deeper than the methods
used. Methods are simply a vehicle in which the researcher can
travel from curiosity to theory. They alone do not define qualitative
research.
METHODOLOGY
Given the underlying tenet of this investigation that knowledge
is socially constructed through interactions with various mediating
agents, it was necessary to look within the various forums in which a
prospective teacher’s pedagogical content knowledge is mediated.
These include the mathematics classroom assigned to the
prospective teacher, meetings between the prospective teacher and
the university supervisor, as well as opportunities for reflection by
the prospective teacher. Other forums exist, such as the prospective
teacher’s meetings with peers or the cooperating teacher. However,
this study focused on one prospective teacher’s interactions with her
students and the university supervisor.
It should be noted that, although the prospective teacher’s
students would not typically be viewed as that teacher’s more
knowing others in terms of mathematical content, they are more
knowing others with respect to existing classroom norms. As such,
they will eventually generate contexts in which negotiation with the
teacher is required in order to achieve a taken-as-shared basis for
communicating mathematics in the classroom. The mediation of
pedagogical content knowledge occurring as a result of this was of
interest here.
Methodological Framework
A naturalistic mode of inquiry was adopted to address the
questions of this study. In particular, case studies incorporating
some of the design elements from the constant comparative method
(Glaser & Strauss, 1967) provided the methodological framework.
The constant comparative method can be described as a series of
steps that begins with collecting data and identifying key issues from
the data that become categories of focus. More data are collected to
explore the dimensions of such categories and to describe incidents
associated with them as an explanatory model emerges. The data
and emerging model are then analyzed to understand attendant
social processes and relationships. This is followed by a process of
coding and writing as the analysis focuses on core categories. The
entire process is repeated continuously throughout the data
collection as developing themes are refined (Bogdan & Biklen,
1992). The resulting explanation of the phenomenon under
investigation is often characterized as grounded theory in that it
emerges inductively from the data.
Here, the case studies of prospective middle school
mathematics teachers were treated as microethnographies. That is,
the studies were characterized by a sociocultural interpretation of
the data (Merriam, 1988), with the added assumption that each of
the prospective teachers’ classrooms would develop unique practices
for doing and talking about mathematics and mathematics teaching
(Underwood-Gregg, 1995). Additionally, the task of understanding
prospective teachers’ constructions of pedagogical content
knowledge during the professional semester called for a teaching
experiment. This was envisioned as an extension of Steffe’s (1991)
use of a constructivist teaching experiment to elicit models of
children’s mathematical constructions. In particular, the prospective
teacher, as student, was constructing pedagogical content
knowledge. The university supervisor, as teacher, assisted through
instruction.
Participants
Three prospective middle school mathematics teachers in their
final year of a four-year teacher education program at a large
southeastern university agreed to participate in this study. All three
had selected mathematics as an area of concentration; two had
opted for a dual concentration in mathematics and science. All were
members of a cohort of 47 students participating in an ongoing
investigation of the sociocultural mediators of learning during their
professional semester. The participants’ membership in this cohort
allowed the researcher increased accessibility to their mathematics
classrooms and, as such, was used as a selection criterion. The
participants, ranging in age from 21 to 24, included one European-
American female, one African-American male, and one European-
American male. They were selected to reflect diversity with respect
to race and gender. Additionally, all had average to above average
university academic experiences and were expected to successfully
complete their student teaching practicum.
Data Collection
The methodological framework of this study necessarily guided
the data collection. In particular, multiple methods appropriate
within a qualitative paradigm were used to collect data. Such
methods included participant observation, in-depth interviews, and
artifact reviews. In particular, the university supervisor observed
each of the three prospective teachers one day per week during two
different sections of a selected course for the twelve-week student
teaching practicum. During each visit, the prospective teacher
participated in a teaching episode interview. The observations were
planned by a telephone conference with the prospective teacher
prior to each visit. Field notes taken during the observations focused
on teacher-student interactions which indicated the prospective
teacher’s pedagogical content knowledge.
Episodes of discourse in the prospective teacher’s
mathematics classroom reflecting mediation of that teacher’s
pedagogical content knowledge became the focus of in-depth
interviews between the university supervisor and the prospective
teacher. In particular, the 45-minute interviews were used as
teaching episodes to further mediate the prospective teacher’s ideas
about teaching mathematics. New understanding resulting from the
episodes were used to generate alternative instructional strategies
for subsequent classes.
When teaching schedules permitted, the interview took place
between successive observations of same-subject instruction so as to
provide interventive mediation. Otherwise, it was scheduled after
the two classroom observations had occurred. Interview protocols
were modified as the study progressed to reflect the direction of the
data. All classroom observations and interviews were audiotaped and
videotaped. Finally, the participants were asked to write personal
reflections on mediation that occurred in classroom and interview
episodes of discourse.
The supervisory process of observation, teaching episode,
observation, and written reflection that the prospective teachers
experienced as part of this study is described here as the cycle of
mediation (see Figure 2). It is seen as cyclic in that new knowledge
about teaching mathematics should be reflected in future lessons as
the teacher’s practice emerges.
Other written artifacts including participants’ lesson plans and
related instructional materials, as well as teaching portfolios, were
included in the data corpus. Additionally, I audiotaped reflections
immediately following each visit in order to record my impressions
and ideas. Furthermore, each cooperating teacher was interviewed
twice during the practicum to obtain a more global picture of the
student teacher’s social context. Documents such as interview
protocols and consent forms necessary for the execution of this study
are included in the appendix.
Figure 2. The cycle of mediation in an emerging practice of teaching.
Data Analysis
The descriptive data corpus generated in this study was
analyzed inductively for themes emerging throughout the process of
data collection and as a result of working with the collected data.
Analysis in a qualitative research study is a systematic process of
sense-making that begins in the field (i. e., the place of data
collection). At this point, the purpose is to narrow the focus of the
study, to refine research questions, and plan sessions of data
collection in light of emerging themes. In this study, issues
concerning the prospective teacher’s pedagogical content knowledge
arising within episodes of discourse in the mathematics classroom
served to narrow the focus of inquiry during the data collection.
Given the dynamic process of becoming a teacher, it was expected
that the focus of research with each of the three participants would
be different. This, coupled with the extensive data corpus generated
by the study, required selecting one of the prospective teachers for
complete analysis after data collection. Hereafter, I will refer to that
participant as Mary Ann (pseudonym).
The analysis that occurred after the data had been collected
involved arranging the data into manageable pieces in order to
search for patterns, discover what was important, and decide what
to tell others (Bogdan & Biklen, 1992). This is often described by
qualitative researchers as finding the story in the data. To
accomplish this, transcripts from the audiovisual recordings of
observations and interviews with Mary Ann were reviewed for
episodes of meaningful interactions between Mary Ann and her
students or her university supervisor. Such episodes were noted and
further analyzed for the mediating role of conversation, or discourse,
in learning to teach mathematics. From this, appropriate segments
were selected for further analysis. Additionally, written artifacts (e.
g., journal reflections) supplementing these data were combed for
confirming or disconfirming evidence of assertions about Mary Ann’s
pedagogical content knowledge. Coding categories developed from
the analysis were refined through multiple sorts of the data. The
data were then analyzed longitudinally to determine how Mary Ann’s
ideas about teaching mathematics developed during the professional
semester as a result of social interactions. The process of analysis as
it relates to the specific questions of this study is outlined more
extensively in Part III and Part IV.
Role of the Researcher
A hermeneutical approach to research is subjective in that the
researcher, by choice, is situated within the context of the
investigation. As such, it is necessary here to discuss my role in this
investigation. In particular, I was both investigator of the study as
well as the university supervisor for the prospective teachers. While
this dual function of nonjudgmental observer and university
evaluator may seem incongruous, it served to minimize my intrusions
into the prospective teachers’ mathematics classrooms. This was
ultimately the greater priority, given the many challenges
prospective teachers already face during their practicum.
One of the advantages of this dual role is that it offered an
inside perspective from which to study the process of becoming a
mathematics teacher. Rather than doing research on prospective
teachers, I was involved in a collaborative effort with them to
improve their mathematics teaching. This view of teachers as
collaborators in research has become the norm as scholars recognize
the necessity of the teacher’s voice (Shulman, 1992). Others (e. g.,
Ball, 1993; Lampert, 1992) have used a similar approach in their
research by becoming teachers in the mathematics classroom.
In an analogous manner, I became the teacher for the
participants in a classroom where mathematics pedagogy was the
content. This allowed me to use instruction to create a zone of
proximal development for the prospective teachers during the cycle
of mediation. In this sense, I became the adult or more capable peer,
as conceived by Vygotsky (1986), for the prospective teachers.
MATHEMATICAL DISCOURSE IN A PROSPECTIVE TEACHER’S
CLASSROOM: THE CASE OF A DEVELOPING PRACTICE
Maria L. Blanton
North Carolina State University
Abstract
This investigation is a microethnographic study of a
prospective middle school mathematics teacher’s emerging practice
during the professional semester. In particular, a Vygotskian (1986)
sociocultural perspective on learning is assumed to examine the
nature of classroom discourse and its role in a teacher’s construction
of pedagogical content knowledge.
Classroom observations, teaching episode interviews, and
artifact reviews were used to document the practice of Mary Ann
(pseudonym) during the student teaching practicum. From the data
corpus, mathematical discourse embedded in classroom interactions
was analyzed with respect to pattern and function. Analysis of early
classroom interactions indicated that students’ awareness of
classroom norms for doing mathematics positioned them as Mary
Ann’s more knowing others, thereby contributing to a reciprocal
affirmation of the traditional roles of teacher and student. Moreover,
discourse seemed to play a dialectical role in Mary Ann’s
construction of pedagogical content knowledge, as her obligations in
the classroom transitioned from funneling students to her
interpretation of a problem to arbitrating students’ ideas.
The influence of Mary Ann’s interactions with her students on
her understanding of how to teach mathematics presents a challenge
to teacher educators to help teachers develop their craft in the
context of the classroom.
Introduction
In recent years, the preeminence of constructivism as an
epistemological orientation in mathematics education has directed
much attention toward understanding how students construct
mathematical knowledge (e. g., Bartolini-Bussi, 1991; Cobb 1995;
Cobb, Yackel, & Wood, 1992; Lo, Wheatley, & Smith, 1991; Steffe &
Tzur, 1994; Thompson, 1994). This focus has often led to interpretive
inquiries into classroom discourse as researchers seek to explicate
the nature of students’ mathematical thinking (e. g., Cobb, 1995;
Cobb, Boufi, McClain, & Whitenack, 1997). Since the National
Council of Teachers of Mathematics (NCTM) Curriculum and
Evaluation Standards for School Mathematics (1989) has prioritized
classroom communication as a facilitator of students’ mathematical
understanding, an ongoing research interest in discourse seems
assured. Indeed, a continued emphasis on classroom discourse is
pivotal to current reforms in mathematics education because it
informs not only our understanding of students’ thinking about
mathematics, but also teachers’ thinking about teaching
mathematics. Recent studies in the professional development of
mathematics teachers (e. g., Cobb, Yackel, & Wood, 1991; Peressini
& Knuth, in press; Wood, 1994; Wood, Cobb, & Yackel, 1991) have
broadened our vision of classroom discourse as a catalyst for teacher
learning. Cobb, Yackel, and Wood (1991) maintain that “it is as
teachers interact with their students in concrete situations that they
encounter problems that call for reflection and deliberation. These
are the occasions where teachers learn from experience (p. 90).”
However, the nature of classroom discourse and its concomitant role
in a teacher’s construction of pedagogical content knowledge is still
underdeveloped.
Wood (1995) addresses this deficit in the literature with an
interactional analysis of classroom discourse that situates the
teacher as the learner. In her study, classroom discourse is valued as
giving voice to the social complexities inherent in teaching in a
collective setting. By documenting patterns of interaction between
teacher and students as they negotiate their roles in the classroom,
discourse provides a verbal window into the teacher’s developing
practice. This genre of research on teacher development in situ
suggests an interesting parallel for the study of prospective teachers
during the professional semester, that is, the student teaching
practicum. Until this time, prospective teachers’ understanding of
how to teach mathematics is almost necessarily academic.
Prospective teachers may be primarily confined to university settings
which offer only decontextualized opportunities for developing their
craft. The professional semester offers the optimal context in which
knowledge of mathematics and mathematics teaching and learning
coalesce into an emerging practice for the neophyte teacher. Here,
my curiosity centers on the role discourse plays in this process.
Specifically, this study is guided by the following research questions:
1. What is the nature of mathematical discourse in a
prospective teacher’s classroom?
2. What does such discourse suggest about the prospective
teacher’s pedagogical content knowledge?
3. How is the prospective teacher’s pedagogical content
knowledge mediated through such discourse?
Since the notion of classroom discourse connotes a variety of
meanings, I specify it here to denote talk, or utterances, about
mathematics made by teacher and students in the classroom.
Teacher Learning Through Classroom Discourse
Vygotsky’s (1986) sociocultural approach gives theoretical
precedent to the place of discourse in an individual’s development.
According to Minick (1996), Vygotsky maintained that “higher
voluntary forms of human behavior have their roots in social
interaction, in the individual’s participation in social behaviors that
are mediated by speech [italics added]” (p. 33). Vygotsky extends
this idea in his general genetic law of cultural development, which
posits that an individual’s higher mental functioning appears first on
the intermental plane, between people, and is then genetically
transformed to the intramental plane within the individual. The
significance of this perspective is that it extinguishes traditional
boundaries between individual and social processes in order to forge
a view of mind constituted by both (Wertsch & Toma, 1995). Bateson
succinctly illustrates this notion of an extended mental system:
Suppose I am a blind man, and I use a stick. I go tap, tap, tap.
Where do I start? Is my mental system bounded at the hand of
the stick? Is it bounded by my skin? Does it start halfway up
the stick? Does it start at the tip of my stick? (Bateson, 1972,
as cited in Cole & Wertsch, 1994).
Therefore, Vygotsky’s belief in the social origins of higher mental
functioning embeds human consciousness in “the external processes
of social life, in the social and historical forms of human existence”
(Luria, 1981, as cited in Wertsch & Tulviste, 1996, p. 54). In the
external processes of the classroom setting, the teacher is also
subject to this social formation of mind. That is, the teacher’s
obligation to manage the intermental context of the classroom
generates opportunities for that teacher to learn as well. The activity
of teaching, of deciding what mathematical knowledge students need
and when meaning has been constructed, continually creates
dilemmas for the teacher to resolve in the process of classroom
instruction (Wood, 1995). Thus, understanding a teacher’s
construction of knowledge about teaching mathematics is inherently
linked to the social dynamics of the classroom.
Although Vygotsky theorized that higher mental functioning is
mediated by both physical and psychological socioculturally-evolved
tools (Wertsch, 1988), it was his belief in the primacy of language as
a mediating tool that drew my attention to classroom discourse.
Concerning language, Vygotsky further reasoned that, as a higher
mental function, language is itself subject to mediation. Holzman
(1996) explains this seeming conundrum:
The dialectical role of speech is that it plays a part in defining
the task setting; this activity redefines the situation, and in
turn, speech is redefined. Language is both tool and result of
interpersonal [i. e., intermental] and intrapersonal [i. e.,
intramental] psychological functioning (p. 91).
Such dualism lends further support to the centrality of discourse in a
teacher’s developing practice. That is to say, in the intermental
context of the classroom, it is primarily discourse, or the language
embedded therein, that mediates the teacher’s practice.
Furthermore, the nature of such discourse is a harbinger of the
teacher’s internalized thinking about teaching mathematics. Under
the umbrella of Vygotsky’s general genetic law of cultural
development, Wertsch and Toma (1995) maintain that the nature of
classroom discourse induces an active or passive stance on the part
of the student, which is subsequently echoed in that student’s
intramental functioning. This principle concerning the relationship
between one’s external and internal speech can be extended to the
teacher as well. In other words, the nature of classroom discourse
will be reflected in the teacher’s intramental thinking about teaching
mathematics. Finally, the effect of speech being redefined through
social interactions is then reflected in an emergent form of
languaging by the teacher. Therefore, language is central in a
cyclical process of development through which it mediates higher
mental functioning first intermentally, then intramentally. As
language voices that mediated higher mental functioning, the
process is renewed.
As an illustration, consider a teacher’s attempt to help a
student resolve a mathematical dilemma. In the process of discourse,
the teacher attempts to make sense of the student’s difficulty and
decides on a course of action. As the instructional plan unfolds, the
teacher tries to assess the student’s understanding and may
subsequently modify the plan in order to influence that student’s
thinking in a desired direction. In effect, the teacher’s behavior (as
well as the student’s) is being mediated in the context of this
interaction. What emerges for the teacher is a new awareness of
how to address a student’s difficulty at some level of generality, an
awareness that is reflected through variations in the teacher’s
speech. The teacher’s practice should increasingly reflect a depth of
experience born out of interactions with students.
Process Of Inquiry
I adopted an interpretive approach (Erickson, 1986) to
consider the developing practice of Mary Ann (pseudonym), a
prospective middle school science and mathematics teacher. Mary
Ann was in her final year of a four-year teacher education program
when asked to participate in this study. From our first meeting in
which I explained the purpose of my research, the professional
contribution that she could make, and my role as her university
supervisor, Mary Ann’s enthusiasm promised a partnership from
which we both could learn.
The Research Setting
I treated the case study of Mary Ann as a microethnography.
That is, viewing the classroom as a socially and culturally organized
setting, I was interested in the meanings that teacher and student
brought to discourse and how this shaped the teacher’s practice
(Erickson, 1986). Since such an approach presumes that classrooms
will develop as separate microcultures, I introduce the reader here
to the school community into which Mary Ann was acculturated as a
student teacher.
The county in which Mary Ann was assigned a student
teaching position is situated in a large urban area that supports 19
public middle schools, enrolling about 20,000 sixth-, seventh-, and
eighth-grade students. Mary Ann’s assigned school reflected a
relatively diverse student population of 1200. Progressive discipline,
site-based management, and the cooperation of parents and
community were hallmarks of its infrastructure. Outside of the
classroom, teachers worked in interdisciplinary teams to integrate
the various content areas. Within this system, Mary Ann was
assigned to a seventh-grade mathematics classroom in which she
taught general mathematics and pre-algebra. She was paired with a
cooperating teacher who provided a nurturing atmosphere for Mary
Ann.
Collecting the Data
Although my focus here is on discourse in the prospective
teacher’s classroom, the data corpus reflects broader issues in Mary
Ann’s developing practice. Specifically, participant observation, in-
depth interviews, and artifact reviews were selected as tools of
inquiry. Weekly visits with Mary Ann during the practicum were a
three-hour interval that consisted of a classroom observation,
followed immediately by a teaching episode interview, and finally, a
second classroom observation. Both observations were of Mary Ann
teaching general mathematics. Each visit was documented through
field notes and audio and audiovisual recordings.
Mary Ann was also asked to provide a copy of her lesson plan
along with any supporting materials, such as quizzes or activity
sheets, at each visit. Although these documents were viewed as
secondary data sources, I could not assume that key issues might not
later emerge from them. Additionally, Mary Ann was asked to keep a
personal journal in which she reflected on what she had learned
about her students, about mathematics, and about teaching
mathematics through the course of each visit. After each visit, I
audiotaped personal reflections about emerging pedagogical content
issues and how future visits could incorporate these themes as
learning opportunities for Mary Ann. In all, I had eight visits with
Mary Ann, followed by a separate exit interview. Finally, I conducted
two clinical interviews with the cooperating teacher to obtain a more
complete picture of Mary Ann’s classroom community (see
Appendix).
Analyzing Classroom Discourse
Pattern And Function In Teacher-Student Talk
I have outlined a process of data collection that is inclusive of
multiple influences in a teacher’s development. To examine the
questions posed in this study about classroom discourse, I focused
on classroom observations as the primary data source. Having
previously established the theoretical motivation for an analysis of
classroom discourse as a window into the student teacher’s
developing practice, I now turn to the specifics of such an analysis.
Discourse analysis rests upon the “details of passages of discourse,
however fragmented and contradictory, and with what is actually
said or written” (Potter & Wetherell, 1987, p. 168). The tendency to
read for gist, or to reconstruct the meaning in someone’s words so
that it makes sense to the reader or listener, should be resisted.
Because such an analysis is often tedious and unscripted, I have
attempted to concisely delineate that process here.
According to Potter and Wetherell (1987), there are essentially
two phases in discourse analysis: (1) identifying patterns of
variability and consistency in the data, and (2) establishing the
functions and effects of people’s talk. Pattern and function captured
the nature of discourse in Mary Ann’s classroom and thereby
revealed the essence of her developing knowledge about teaching
mathematics. Furthermore, based on Wood’s (1995) process of
documenting teacher learning in the classroom, I looked at shifts in
pattern and function to establish Mary Ann’s construction of
pedagogical content knowledge.
Current literature (e. g., Underwood-Gregg, 1995; Wood,
1995) provided insight into identifying patterns in classroom
discourse. Speaking from the traditions of ethnomethodology and
symbolic interactionism, Underwood-Gregg explains that obligations
felt by teacher and students in accordance with their perceived roles
in the classroom are enacted through various routines. Such
routines, most often embedded in language, comprise the patterns of
interaction in the classroom. For example , Mary Ann’s felt
obligation to clarify a student’s thinking was often enacted as a
routine in which she asked a series of instructional questions (i. e.,
those for which the teacher already knows the answer [Wertsch &
Toma, 1995]) designed to lead that student, step-by-step, to the
correct solution. Simultaneously, the student’s obligation to give the
teacher’s desired response sometimes led to a routine of guessing by
that student. Together, these routines comprised a pattern of
classroom interaction. Thus, identifying a pattern in the data
requires constructing its constituent parts, namely, the routines of
teacher and students that give rise to that pattern.
Identifying the function of discourse in the classroom leads to
a myriad of nuances in the teacher’s utterances which, in aggregate,
give voice to her mathematics pedagogy. Thus, drawing from the
work of Wertsch and Toma (1995), I appealed to Soviet semiotician
Yuri Lotman’s (1988) dichotomy of the function of text as univocal or
dialogic to provide a clarifying lens on this aspect of discourse.
Lotman broadly defines text as a “semiotic space in which languages
interfere, interact, and organize themselves hierarchically” (p. 37).
This includes written words, verbal utterances, and even art forms.
By univocal functioning, Lotman implies text that serves as a
“passive link in conveying some constant information between input
(sender) and output (receiver)” (p. 36).
As an illustration, consider teacher-student interactions in
which the teacher asks a series of instructional questions. In this
case, neither teacher nor student needs to actively participate.
Moreover, any discrepancy between what is transmitted and what is
received is attributed to a breakdown in communication. In contrast,
dialogic functioning refers to text that is taken as a “thinking
device”. That is, rather than being interpreted as an encoded
message to be accurately received, the speaker’s utterances serve to
generate new meaning for the respondent, who takes an active
stance toward the utterance by questioning, validating, or even
rejecting it (Wertsch & Toma, 1995). As such, it is likely that
students initiating and maintaining dialogic interactions may run
counter to typical (American) classroom norms, thereby making it
the responsibility of teachers and teacher educators to cultivate
dialogic functioning in the intermental context of the classroom.
Process of Analysis
Teasing out pattern and function from discourse data seemed
arduous at the outset. I began by transcribing audiovisual recordings
of classroom observations, inserting comments and questions as they
arose in transcription. In retrospect, these memorandums initiated
my sense-making of the data corpus. Using the conversational turn
as the basic unit of analysis, I combed the early transcripts to
identify a preliminary coding scheme that would describe the
purpose of Mary Ann’s utterances. For example, her questions
“What’s the common denominator between six and two?” and “How
did you figure out that six was the common denominator?” were
coded as “Request for Computation” [RFC] and “Request for
Procedure” [RFP], respectively. Such codes reflected Mary Ann’s
expectations of students as participants in mathematical discourse,
thereby providing insight into her thinking about teaching
mathematics. From this preliminary scheme, codes were refined or
discarded and new codes were added as subsequent data were
analyzed. (See Appendix for this coding scheme.)
To code the transcripts, each classroom observation was
divided into manageable sections based on naturally occurring
divisions in the sequence of classroom events. Such divisions were
signaled by a change in theme or direction, such as the conclusion of
class discussion on a particular problem. Sections were then coded
by conversational turn and the essence of interactions between Mary
Ann and her students was abstracted to get a sense of the routines
and patterns in the discourse. Additionally, sections were compared
in order to ascertain similarities and differences that suggested
changes in Mary Ann’s practice. The coding system represented my
first attempt at sorting the data and was eventually set aside as I
focused on the particulars of pattern and function in the discourse.
Once all of the transcripts had been coded, four classroom
observations representative of Mary Ann’s developing practice were
selected for further analysis. In deference to the cultural personality
intrinsic to individual classes, I chose all of these observations from
Mary Ann’s third period general mathematics class. Based on the
work of Underwood-Gregg (1995) and my own preliminary analysis, I
considered the routine actions that Mary Ann and her students
enacted subsequent to the following interdependent events:
a student posed a mathematical question
a student responded to a mathematical question
the teacher posed a mathematical question
the teacher responded to a mathematical question
From the four classroom observations, sections were selected as
representative of the routines and patterns manifested following
these events. These sections were then analyzed to characterize the
function of text as univocal or dialogic. Since function is identified by
the respondent’s passive or active interpretation of the speaker’s
utterance, it was necessary to look at each speaker’s utterance and
how it was subsequently interpreted (e. g., as a thinking device) by
the respondent. Additionally, I met periodically with my advisors and
other available faculty and graduate students to review the
audiovisual recordings and discuss the nature of discourse in Mary
Ann’s classroom, what it suggested about her pedagogical content
knowledge, and how it mediated that knowledge. Other data sources
(e. g., written artifacts) were perused for confirming or
disconfirming evidence concerning assertions generated through the
analysis.
Findings and Interpretations
Early Pattern and Function in Classroom Discourse
In this section, I discuss through transcription and analysis the
nature of early discourse in Mary Ann’s classroom and what such
discourse suggested about her pedagogical content knowledge while
in its infancy. Mary Ann’s early practice metaphorically identified
her as the captain of a ship, keenly obligated to navigate rough
waters for her students. Taking over the helm of the classroom when
all sailing seemed smooth only intensified her need to ensure
students’ cognitive calm. As Mary Ann anticipated mathematical
storms for her students, she often rushed to avert them by giving
information and explaining procedures, or changing the problem in
question altogether. As the captain, it was primarily her place to do
this. Indeed, she became the hero by skirting the hazards of
unknown waters. While this was a commendable role for Mary Ann,
it sometimes hindered students from steering themselves, as they
yielded the balance of responsibility to her.
Early pattern and function in resolving students’ mathematical
dilemmas. Throughout the practicum, Mary Ann’s usual custom was
to begin class with students’ questions from the previous night’s
homework, introduce new topics, and then close the lesson with
practice problems or a short quiz. The following excerpt from the
transcripts typifies the manner in which she addressed students’
mathematical questions during the early stages of her practice. In
this particular episode, a student (Allyson) has asked Mary Ann
about an exercise from homework. As was often the case when
working problems through whole-class discussion, Mary Ann copied
the exercise on the overhead projector [OP] and recorded
mathematical pieces of the ensuing discussion as students spoke.
(All names are pseudonyms.)
1 Teacher: O. K., what was the first step we want to do,
Allyson?
2 Allyson: Make it a zero?
3 Teacher: O. K., what’s the very first thing? What’s the very
first step yesterday? What did we want to do with that
variable?
4 Allyson: Isolate it.
5 Teacher: Isolate, and I want everybody to start using this
term, “isolate”. It’s a mathematical, algebra term
and I want you to learn how to use it. O. K., I know
you’re not used to seeing the variable on this side (right
side), so if you want to rewrite it, and just switch, you
can just switch it around like this (Mary Ann illustrates
on the OP.). That’s the same thing. O. K., so now we
want to isolate the variable, but what have we got to
do before we isolate the variable? (A student
indicates that they should evaluate the exponent.)
O. K., we want to get rid of that exponent. So what is
nine squared?
6 Students: Eighty-one. (One student says eighteen.)
7 Teacher: Who said eighteen? How did you get eighteen? I’d
like to know.
8 Student: I was thinking nine times two.
9 Teacher: O. K., remember that when you see nine squared,
that’s not nine times the exponent. That’s nine
times itself, and in this case you write nine down
twice. O. K., so then you’ve got one hundred and
twenty-one. O. K., so now how do we isolate the
variable? (Allyson’s response is inaudible to me.) O.
K., so subtract eighty-one, and I want you to start
using the term. When I ask, “How do you isolate the
variable?”, you say, “Subtract eighty-one from both
sides”. So you don’t have to say, “Subtract it from this
side, then subtract it over here”. Just tell me you
subtract eighty-one from both sides. O. K., so eighty- one
minus eighty-one?
10 Allyson: Zero.
11 Teacher: Zero. O. K., we have to line up the decimals, right?
So there’s an understood decimal behind eighty-one.
So five minus zero?
12 Allyson: Five.
13 Teacher: One minus one?
14 Allyson: Zero.
15 Teacher: Now we have to borrow, so that becomes zero
because we borrowed a whole. (Mary Ann pauses to
get the attention of several students who have
started to talk with each other.) Eight from twelve?
16 Allyson: Four.
17 Teacher: O. K., so you just have s equals...(her voice trails
off as she writes the final answer on the OP.)
The appearance of the correct solution signaled an end to the
episode and Mary Ann moved on to the next question.
Mary Ann began the dialogue outlined above by establishing
her approach for working the exercise, supplying Allyson with non-
mathematical, referent-laden hints that would prompt Allyson’s
recall of the procedure she needed to follow (1, 3). Allyson’s
unsuccessful attempt (2) to give the response that Mary Ann wanted
prompted Mary Ann to enact a “giving hints routine” (3). Allyson’s
obligation in this interaction was to guess the desired response,
upon which Mary Ann could move to the next phase. At this point,
Mary Ann initiated an “incremental questioning routine” in which
she asked a series of cognitively-small, closed, leading questions,
sometimes accompanied by her explanation, that funnelled Allyson
toward a final solution. To her credit, Mary Ann genuinely wanted
students to participate in the process of working the exercise.
However, at this point in her practice, she relied on questioning
strategies that required students primarily to compute simple
answers, recall information, or describe procedures previously
learned (e. g., What have we got to do before we isolate the
variable?, So what is nine squared?, [What is] eight from twelve?).
This type of question-and-answer interaction evoked a vertical
discourse between teacher and student that, given students’ willing
participation, quickly became a classroom norm for doing
mathematics.
The early pattern of interaction constituted by the routines of
Mary Ann and her students that unfolded when a student posed a
mathematical problem is summarized below:
Typical Early Pattern of Interaction
teacher writes the problem/exercise on the OP and sets the
direction for solving the problem by giving information and
asking leading questions
student guesses a response
teacher gives hints in order to get a particular response from
student(s)
student gives desired response
teacher repeats student’s response and asks a leading, follow-
up question.
With the exception of the first step, a variation of this pattern
typically repeated until a correct solution appeared.
This episode between Mary Ann and her students seemed to
indicate a predominantly univocal functioning of text. For example,
Allyson’s incorrect response (2) led Mary Ann to assume that her
original question (1) was either inaccurately transmitted or received.
This signaled Mary Ann to retransmit the message with more
accuracy, that is, give more suggestive hints (3). Allyson’s correct
response (4) then suggested that the message had been accurately
received and Mary Ann could continue (5). As Mary Ann
concentrated on demonstrating her thinking (e. g., 9, 11, 15), she
peppered her explanations with questions that served to check
accuracy in transmission (e. g., 13). Neither Mary Ann nor her
students seemed to treat a speaker’s utterance as something to be
questioned for the purpose of generating new thinking. In other
words, a respondent’s passive interpretation of a speaker’s
utterance designated the function of that utterance as univocal.
Although Mary Ann did question one student’s response (6) in a
seemingly dialogic fashion (7), her purpose was to dispel discrepant
thinking (9).
The obligation that Mary Ann felt to clarify Allyson’s thinking
positioned Mary Ann as the filter of discourse. That is, Mary Ann
initiated the exchange, decided what type of questions to ask, when
and to whom to ask these questions, and when an answer was
acceptable. The norm was for students to respond to the teacher’s
questions, not one another’s ideas. Orchestrating all of this is quite a
challenge, especially for the novice teacher. Although Mary Ann
seemed quite adept, the risk was in her controlling the discourse as
if somehow students were marionettes and she their puppeteer.
Rather than exploring students’ thinking, their ideas and strategies,
Mary Ann was intent on showing how she would have worked the
problem, fishing for student responses that would support her
interpretation. At this early stage in her practice, it seemed inherent
in her beliefs about teaching to be the center of information for her
students, weeding out responses that did not follow a teacher-
selected path for solving the problem at hand. As did all of her
efforts, this approach stemmed from an earnest desire to be a good
teacher.
Early pattern and function in teaching a new concept. On my
second visit with Mary Ann, I observed her teaching a lesson on
adding and subtracting algebraic expressions. I have included
lengthy transcripts from this lesson in order to preserve its integrity.
Mary Ann often tried to motivate new topics with a mathematical
activity that would pique students’ interest. Her opening activity for
this particular lesson reflected these efforts.
18 Teacher: (She hands an envelope to Laura.) You be Student
A, but don’t look at this. Hold it down. (She hands
an envelope to Debbie.) You be Student B. O. K., we
have two students, Laura is Student A, Debbie is
student B, and they’re working at a clothing store,
trying to make some extra money.... O. K., Student A
has an envelope that is one day’s pay. O. K.,
Students A and B are working at a clothing store
and they make the same amount of
money...for one day’s work. O. K., Student A has an
envelope that says “one day’s pay”. Student B has an
envelope that says “one day’s pay plus a three dollar
bonus”, so she got a little extra. Can you tell me
how much [Laura] has in her envelope without
looking in the envelope?
19 Laura: No.
20 Teacher: O. K., the amount is hidden, right? Because I won’t
let you open it. O. K., how much does Student B have?
(She looks around for a student who will respond.) O. K.,
what did you say Dianne? (Dianne’s reply is
inaudible to me.)
21 Teacher: O. K., she’s had one day’s pay with a three dollar
bonus. (Mary Ann’s intonation indicates Dianne’s
response was incorrect.) So she has Laura’s pay with a
three dollar bonus, right? (Various students begin
calling out responses.) O. K., so you know that she
has three dollars, so would she have three more
dollars than what Laura has?
22 Dianne: Yes.
23 Teacher: So we know that she has more than what Laura
has, right?
24 Dianne: Yes.
25 Teacher: O. K., Laura has one day’s pay and we know that
Debbie has one day’s pay plus three dollars. (She
begins to write information on the OP.) O. K., (to
Laura) I want you to open up your envelope and see
what you have.
26 Laura: Twenty dollars.
27 Teacher: Twenty dollars. Now Laura has twenty dollars, so
how much does Debbie have?
28 Students: Twenty-three.
29 Teacher: So y’all think she has twenty-three dollars?
30 Students: Yeah.
31 Teacher: So, we said Laura has twenty dollars. According to
what we’ve written here, she’s got twenty. If [Debbie’s]
got three dollars more, she should have twenty-three
dollars. (To Debbie) O. K., you can open your
envelope and see what you have.
32 Student: Yep [sic], she’s got twenty-three dollars.
One could justifiably argue that Mary Ann, not her students,
was the central player in this activity. A quick glance at teacher and
student routines confirms this. To her credit, Mary Ann seemed to
value the use of physical referents such as integer chips, geoboards,
graphing calculators, or her own creations, as a bridge to abstract
ideas. However, her exposition left little room for dialogic
interactions in the classroom.
From this activity, Mary Ann transitioned into the second
phase of her lesson, a review of the defining characteristics of
equations and expressions.
33 Teacher: O. K., what we’re doing today is talking about
expressions in addition and subtraction, and it’s
been a while since we talked about expressions
anyway, so I [wanted to] refresh your memory....
Ahhh, expression and equation, what is an
expression? Everybody just think about it for a
second. John?
34 John: An unfinished problem...
35 Teacher: O. K., we call it a phrase, an unfinished sentence
(she writes this on the OP). O. K., and what do we call
an equation? Does anyone know?
36 Sharon It was...
37 Teacher: Sharon, raise your hand if you want to answer.
(Turning to Kayla) Kayla? (Kayla’s response is
inaudible to me.) O. K., so it was a complete
number sentence. What is the one main difference
that I told you was between an equation and an
expression, John? (He does not know.) I told you one day
I was going to walk in here and I was going to look
like it. It’s the big difference between an equation
and an expression. (John’s response, inaudible to
me, is not what Mary Ann is looking for.) O. K.,
there’s one symbol that makes the difference. (After a
number of students raise their hands, some making
guttural sounds in order to be recognized, Mary Ann
turns to Marta.) Marta? (No response.) O. K.,
Allyson, can you help Marta out?
38 Allyson: Equal sign?
39 Teacher: An equal sign. Can you give me an example of an
equation, Sharon? (Sharon’s response is
inaudible to me.)
40 Teacher: (She writes Sharon’s response on the OP.) O. K.,
Sharon said that was an equation. Would y’all agree
with that? (Students offer mixed responses of “yes” and
“no”.) You wouldn’t agree with that? Why wouldn’t you
agree with that? (Mary Ann turns to one of the students
who disagreed with Sharon’s claim.) It’s a
complete number sentence, with an equal sign.
Maybe you’re thinking maybe if we wrote some things
like this (she writes on the OP). O. K., that’s an equation,
too. One’s numerical and one’s algebraic. Remember
we talked about that. (She turns her attention to the
whole class.) O. K., what would this be (she writes
another example on the OP)? An equation or
expression? Chris?
41 Chris: Uhm...expression?
42 Teacher: Expression. Why is it an expression?
43 Chris: Because it doesn’t have an equal sign.
This predominantly univocal exchange (to which reviewing
content easily lends itself) continued for several more minutes as
Mary Ann prodded students to recall information. It illustrates her
inclination to enact a routine of supplying non-mathematical
referents (e. g., I told you one day I was going to walk in here and I
was going to look like it) until students guessed her answer. Also,
students responded to Mary Ann, not their peers, thereby granting
her the mathematical authority. Mary Ann’s routine of repeating a
student’s correct response (the signal of affirmation), or meeting
incorrect responses with hints, explanations, or a request for peers
to assist, depicts a cultural norm of doing mathematics in her
classroom in which students looked to the teacher, not to
themselves, to explain, justify, or reject their ideas. Although
students’ acquiescence to this norm seemed to reinforce Mary Ann’s
practice, at one point she did begin to shift her obligation onto
students to argue Sharon’s claim and justify their own thinking (40).
When several students rejected Sharon’s claim, Mary Ann’s response
(You wouldn’t agree with that? Why wouldn’t you agree with that?)
seemed to indicate an attempt at dialogic interaction. At this point in
her practice, such an attempt was atypical. Furthermore, as Mary
Ann then tried to anticipate the student’s thinking (i. e., Maybe
you’re thinking if we wrote something like this.), the student was
unable to respond dialogically.
Following the review, Mary Ann led a whole-class discussion in
converting written expressions into symbolic form.
44 Teacher: O. K., it says (Mary Ann reads from the textbook),
“Jody is entering the pumpkin stacking contest at the
Pumpkin Festival. She’s hoping to balance three more
pumpkins in her stack this year than she did last year.”
So that’s kind of what we were just talking about.
Debbie had three more dollars than Laura did. O.
K.? So she wants more, three more, pumpkins this year.
(Mary Ann writes the problem on the OP.) O. K.,
we’re going to set this up in equation form using a
variable. Remember we talked about a variable?
We’re going to let n equal the number of pumpkins last
year. So we don’t know how many, so we’re just going
to give it a variable. We could have called that t or s
or a or b, whichever variable we want to call it. So she
wants three more. So when we think of more, do we
think of addition or subtraction?
45 Students: Addition.
46 Teacher: Addition. You’re going to add some things on. So
we know we’re going to add, and we know we want
three. It’s just like when we said we want one day’s
pay plus three dollars is what Debbie had. So this is
the number of pumpkins she had last year, plus the
three more she wants this year.
It is interesting to note that in this episode’s entirety, students
were asked only to determine if “more” implied addition or
subtraction. This underscores a recurring theme of univocal
discourse that positioned Mary Ann as the sender and students as
receivers of information. She continued this pattern of interaction
with a series of related tasks whereby, for each task, she read a
written or algebraic expression (e. g., a plus four vis-à-vis a + 4),
then asked instructional questions, sometimes offering explanations
and hints, in order to garner a particular response from students.
The following conversation highlights these interactions and further
supports the assertion that Mary Ann’s knowledge about teaching
mathematics prioritized teacher demonstration as a vehicle for
student learning.
47 Teacher: O. K., how would we say, using more than and
following our pattern, would we say a plus four?
Ron? We want to use our pattern that we have up here
(on the board). A plus four using more than? (Ron’s
response is inaudible to me.) See how I said three
more than n (referring to a previous problem)? I
rewrote that in words. I rewrote n + 3, the
expression n + 3, into words saying three more than n.
So how would I say this right here (i. e., a + 4) in
words using more than? Nunice, can you help
out?
A conversation later in the lesson illustrates what Mary Ann
had intended when she asked a student to solve a particular task.
48 Teacher: O. K., Tom I want you to do (i. e., convert to
symbolic form) the sum of a number z and five. O. K.,
let’s look for what symbol we’re going to use. We said
sum was what? Addition or subtraction?
49 Tom: Addition.
50 Teacher: O. K. Addition (Mary Ann writes a plus sign on the
OP.) O. K., where do you want me to put the z and five?
51 Tom: Z would go on that side (pointing to the left side).
52 Teacher: O. K., and the five would go over here (indicating
the right side)? (Tom nods agreement.)
In this episode, Mary Ann again enacts an incremental
questioning routine in order to funnel Tom to the correct solution. In
her request for Tom to “do the sum of a number z and five” (48),
Tom only had to associate the word “sum” with addition or
subtraction (for which, of course, he had a 50 percent chance of a
correct guess). In her eagerness, Mary Ann genuinely wanted Tom to
be successful. This characteristic of her teaching seems to partially
explain why she fractured the content into cognitively-small, leading
questions. It was as if her responsibility was to help students avoid
any of the struggles that, in reality, do (and should) accompany
mathematical inquiry.
After several more similar episodes, Mary Ann concluded the
lesson with a visual activity on evaluating expressions. She passed
out cards containing either a number, variable, or mathematical
symbol, to volunteers who had not participated on this particular
day. As she read a written expression (e. g., five more than s) aloud,
students with the corresponding parts (i. e., 5, +, and s) arranged
themselves at the front of the room. Occasionally prompted by Mary
Ann, they held their cards to indicate the expression s + 5.
53 Teacher: Now we’re going to work this out and find a value,
so I need whoever is going to make this sentence
complete and make it into an equation [to] come up
here. (The student with the “=“ card walks to the front.)
O. K., I want s to equal four. (The student with the
“4” card walks to the front.) I want to bump...s
and put four in. (Speaking to the student with the “s”
card) So you stand behind her (indicating the
student with the “4” card). I’m replacing the variable
s with the number four. Now we’ve got to find the
value, so whoever thinks they have the answer to
this, come on up. (The student with the “9” card walks
to the front.) All right. Very good. So what we did
was we replaced s, our variable. We bumped her
(indicating the student with the “s” card) and put in four.
We made a what? An expression or equation?
54 Students: Equation.
As with the opening activity of this lesson, Mary Ann again
purposed to situate an abstract idea in a concrete setting, this time
using students as visual referents to personify evaluating
expressions. Also as before, she assumed the responsibility of
explaining the process as well as the conclusions, leaving students
with only minimal input. Even so, this activity’s inclusion signaled
the importance Mary Ann attached to concrete experiences in
making mathematics meaningful for students.
On early discourse and Mary Ann’s practice. The early pattern
in classroom interactions that unfolded when Mary Ann taught new
concepts was equivalent in structure to the pattern exhibited when
she addressed students’ homework questions, outlined earlier in this
section. That is, whether Mary Ann or a student asked a question or
posed a task to be solved, Mary Ann typically established the
solution approach by giving information, asking leading questions, or
both (cf. 1, 44), then proceeded to direct students to the correct
solution through questions and hints (cf. 3-5, 37-39). Moreover, the
self-perceived roles of teacher and students in mathematical
discourse, manifested through their routine actions, led almost
exclusively to univocal classroom interactions.
What I observed in these early patterns of discourse is not
unlike those outlined elsewhere in the literature. In what Bauersfeld
(1988) describes as a funnel pattern, the teacher asks questions to
which he or she already has an answer. If a student gives an
incorrect response, the teacher then tells the correct response or
directs the student step-by-step to the correct answer. Underwood-
Gregg (1995) describes what Voigt has identified as an elicitation
pattern. In this, the teacher vaguely poses a question for which
students are obligated to offer a variety of answers. The teacher’s
need to direct how the question is to be answered creates the
obligation to follow students’ ideas that match those of the teacher,
or give hints in order to move students toward the teacher’s
thinking.
Such patterns of interaction in the classroom, as well as
discourse that is in essence univocal, have been documented in the
case of inservice mathematics teachers (e. g., Underwood-Gregg,
1995; Peressini & Knuth, in press; Wood, 1995). It is the occurrence
of such discourse from the outset of a prospective teacher’s practice
that is of note here. The student teacher undergoes a cultural
metamorphosis from learner of mathematics to teacher of
mathematics during the professional semester. If that student
teacher’s intramental thinking about mathematics is predominantly
say, univocal, then his or her initial teaching practice would reflect
this. That is, how one teaches mathematics is grounded in how one
thinks about mathematics. Mary Ann’s comments about the role of
problem solving in mathematics during an early teaching episode
identified a consistent link between her thinking about mathematics
and her early practice:
I know that math is one big word problem in itself, because
one thing builds on another. But I don’t look at it like that. I
look at math as just operations you go through, just like a
series of steps. You have to step on this step before you get to
the next one.
In this sense, Mary Ann’s early languaging in the classroom
seemed to be an external representation of her intramental thinking
about mathematics. This, coupled with the claim by Wertsch and
Toma (1995) that 80 percent of American classrooms bequeath
univocality to their students’ intramental thinking about
mathematics, made it likely that univocal discourse would dominate
Mary Ann’s early practice. Moreover, it seemed that the inertia
generated by univocal teacher-student interactions in Mary Ann’s
early practice held implications for her development. This intensified
the need to address her pedagogical content knowledge in its
infancy, in the context of her practice.
Indications of an Emerging Practice: Change in Pattern and Function
The “problem-solving day”. Although the pattern and function
that typified early languaging in Mary Ann’s classroom persisted
throughout the practicum, later discourse did substantiate emerging
patterns in her interactions with students, as well as a shift from
discourse grounded almost exclusively in univocal functioning. My
third visit with Mary Ann, later monikered the “problem-solving day”
because of the lesson’s focus, revealed such changes. I reiterate that
the purpose of the present study is not to address the role of
contexts external to the classroom on changes in Mary Ann’s
practice. Clearly, such contexts (e. g., interactions with the
university supervisor or cooperating teacher) shape the prospective
teacher’s thinking about teaching mathematics, as they did with
Mary Ann. Rather, the purpose here is to explore the nature of
interactions in Mary Ann’s classroom and how those interactions
mediated her pedagogical content knowledge.
The lesson on the problem-solving day dealt with the strategy
“working backwards” as a way to solve simple word problems. Mary
Ann had earlier insisted that she was uncomfortable with word
problems and did not want to teach this particular lesson, yet she
took considerable risks in an attempt to move from her previous
teaching paradigm. After addressing students’ questions from the
homework, she then asked students to work in dyads to solve the
following problem:
Problem 1: I’m thinking of a number that if you divide by three
and then add five, the result is eleven.
Removing herself as the mathematical authority, Mary Ann
seemed to want students to struggle with the problem through peer
interactions and to justify their thinking to one another before she
joined the process. Her attempt to renegotiate classroom norms in
resolving a mathematical question met with immediate resistance
from students as, almost imperceptibly, their role in doing
mathematics had shifted. The following conversation illustrates the
tension created by Mary Ann’s initial efforts to change her practice.
As it begins, a student has just asked Mary Ann if he should write
Problem 1 in his notes.
55 Teacher: If you feel like you need to write it down, write it
down. I just want you to solve it. I’m not going to
answer any questions, just solve it.
A student asked Mary Ann to check her solution. Mary Ann
responded by withholding closure:
56 Teacher: Well, that’s good. You need to write it down and
tell me how you solved it. You should be talking with
your partner. (To the class) Y’all love to talk. Now I’m
letting you talk.
The student again asked Mary Ann to check her solution.
57 Teacher: I’m not going to tell you if it’s right or wrong. I
want you to work it out. You can plug it back in and see
if it’s right.
Another student asked for help, yet Mary Ann continued to resist
intervening. Instead, she encouraged the student to work with her
partner.
58 Teacher: Did you consult with [your partner] and tell her
how you feel about it? (The student indicates she has.)
And she thinks that’s right? (The student again
indicates she has. More students raise their hands.)
No hands up. Just talk about it. (A student tells Mary
Ann she has the answer.) O. K. Good. Then y’all are
ready. (She turns her attention to a particular
dyad.) So have y’all talked about it? You got together?
(She moves to another pair.) Have you figured it out?
(They indicate they have.) And you both agree that this is
your number?
Mary Ann walked around the room several more minutes, stopping
periodically to promote students’ interactions. By the end of this
episode, the classroom resonated with a steady hum as students,
realizing Mary Ann’s intentions, began to communicate
mathematically with each other.
The whole-class discussion that followed reflected another
shift in Mary Ann’s practice, as she pointedly asked different groups
to share their solutions, and later their thinking, with the class.
Noting the first group’s correct response and immediately moving to
others for their solutions, Mary Ann appeared more interested in
understanding students’ thinking than in harvesting only correct
answers.
59 Teacher: O. K., our first group to finish was Debbie and
Susan, so they’re going to tell me the number they got.
(She writes their response “18” on the OP.) O. K.,
(turning to another group), what did you get?
60 Group: Six.
61 Teacher: O. K., what number did you get, Jack?
62 Jack: Eighteen.
63 Teacher: What number did you get (turning to another
group)?
64 Wendy: I got thirty-eight.
At this point, Mary Ann asked Debbie to explain her (correct)
solution of eighteen, to which Debbie responded with a procedural
account of her thinking (65). What seems noteworthy here is that,
by eliciting Debbie’s strategy, Mary Ann was relinquishing a role
which typically she felt obligated to fill.
65 Debbie: It says, ”If you divide by three and add five”, so you
do the opposite. You subtract five from eleven and
that’s six. Then you multiply six times three and
that’s eighteen.
Previously, a correct solution coupled with a correct procedure
would have signaled Mary Ann to repeat that procedure and then
move to the next task. However, in this instance, she turned back to
her students to try to further understand their thinking. After
making sense of the strategies used by those who had found the
correct solution, she asked several groups who had made
unsuccessful attempts to explain their thinking as well, reflecting a
departure from a practice in which she rarely countenanced
incorrect answers. The following episode depicts this.
66 Teacher: (Speaking to another dyad) How did you get
eighteen?
67 Student 1: Same way.
68 Teacher: You did this exact thing?
69: Student 1: No.
At the student’s hesitance to explain his group’s strategy, Mary Ann
turned to another pair frantically waving their hands in order to be
recognized.
70: Teacher: How did you get eighteen?
71: Student 2: We had a number. We said eighteen divided, three
will go into eighteen six times. Then we added five.
72 Teacher: O. K., so first of all you knew that the result had to
be eleven, so you said, “O. K., it’s eleven”. Then I told
you you had added five, so you had to think what added
to five will give you eleven. O. K.? I’m trying to help,
think like you were thinking. Is that what you did?
73 Student 2: Uh huh.
74 Teacher: (To another group) How did you get eighteen?
75 Student 3: Well, we got six.
76 Teacher: O. K., how did you get six?
77 Student 3: O. K., she (indicating her partner) got six because
she just added six to...(Student 3’s partner objects
but her response is inaudible to me). All she did
was added six to five.
78 Teacher: O. K., six to five, but what did you do with the
“three divided by”? (Student 3’s response is
inaudible.) See, it says “three divided by”, so if you
divide three into six, you’re going to get two and two
plus five is seven. O. K., who got the thirty-eight?
I’m curious to see who got thirty-eight. (Student 4
identifies himself.) Tell me how you got thirty-eight.
79 Student 4: I did s over 3.
80 Teacher: You did what?
81 Student 4: S over three.
82 Teacher: O. K., what now?
83 Student 4: Equals eleven.
84 Teacher: So what happened to the five?
85 Student 4: That’s what I said.
As the lesson continued, Mary Ann repeatedly positioned
Debbie as the mathematical authority, thereby allowing Debbie to
retain ownership of her ideas.
86 Teacher: The way Debbie chose to do the problem is what
we’re talking about today. She, well Debbie, you
tell me what you did. Is there any certain way you
can call maybe what you did, without using the book?
When you looked at this problem, where did you
start?
87 Debbie: Where did I start? I started at the answer.
88 Teacher: You started at the answer and then did what?
89 Debbie: And then I just went backwards.
90 Teacher: O. K., did everybody hear what she just said?
Debbie, repeat that one more time.
91 Debbie: I started at the answer and worked backwards and
did the opposite of, uhm, division and addition.
92 Teacher: So Debbie used a problem-solving strategy of
working backwards. That’s just one strategy. Some of
you used guess-and-check, and maybe you didn’t come
up with the right answer, but you were on the right
track. Some of you set up an equation.
Mary Ann’s comments in the interview prior to this class
revealed a different type of thinking about the use of multiple
approaches to solve a problem. Smiling sheepishly, she admitted,
I guess to me, like, I was always, give me a formula, or give me
a way to solve it, and I’ll solve it. And sometimes with word
problems there’s [sic] many different ways.... That puzzles kids
to think there might be more than one way. It always scared
me.... If I know that there...is more than one way, that scares
me. That’s weird, I know, but I feel like if there’s one way, I
can check it, and if I get it right, then I’m right, and I’m right,
and I’m right. That’s all there is to it.
Although she stated her receptiveness to students’ alternative
strategies, Mary Ann’s discomfort with exploring various routes to a
task’s solution was exhibited in early patterns of classroom
interactions where she, not the students, determined the solution
path (e. g., 48-52). That she was now willing to sacrifice the one
strategy she was comfortable with by the inclusion of other valid
processes seemed a significant shift for her.
After posing the following problem to students, Mary Ann once
again turned to Debbie.
Problem 2: The Blueberry Festival is held each Labor Day.
This year there are 89 entries. This is twice the number of last
year’s entries, plus seven. How many entries were in the
blueberry run last year?
93 Teacher: So what were some things when you were working
this other problem that you had to do? O. K., you told us
that you started at the end and you worked to the
beginning. So she started here and she went this way.
But what else did she have to do?
94 Debbie: First I had to write down what, like three divided
by...
95 Teacher: So you said three divided by (she writes this on the
OP). Then what did you say? (Debbie’s response is
inaudible to me.) So you add five, and the result was
eleven. Then what did you have to do? So now you
said you worked from the back end up. So what did you
have to do?
96 Debbie: Then I started with eleven and I subtracted five.
The discussion surrounding Problem 1 and Problem 2 reflects a
variation from typical early languaging in Mary Ann’s classroom. The
whole-class discussion about Problem 1 began with a univocal
sharing of students’ solutions (59-64). This transitioned into students
sharing their strategies (65-89) in a lengthy interaction with Mary
Ann. Furthermore, Mary Ann’s utterances (e. g., 72, 80, 82, 84)
indicated an emerging effort to focus on students’ ideas, not her
own. In particular, she seemed to dialogically question Student 4
(78-84) in order to achieve mutual clarity with him about the
problem’s solution. Previously, she would typically have interpreted
his incorrect solution of thirty-eight as the result of a transmission
error (i. e., univocally) which she was obligated to correct by
demonstrating her own strategy.
Debbie’s utterance (65) later prompted Mary Ann to attempt a
dialogic interaction with her (86-89). I describe this as an attempt
because, although Mary Ann’s questions (e. g., Where did you start?,
You...then did what?) solicited a procedural response from Debbie,
there seemed to be an underlying shift away from instructional
questions to questions that explored Debbie’s thinking.
In Mary Ann’s request, “She, well Debbie, you tell me what you
did”, Mary Ann started to appropriate Debbie’s ideas, then
reconsidered in order to externalize Debbie’s thinking, not
demonstrate her own (86, 88). Mary Ann later continued this
approach in an effort to situate the solution of Problem 2 within the
context of Debbie’s strategy [93-96]. While Debbie’s responses (87,
89, 91) suggested that she still interpreted Mary Ann’s utterances
univocally, I would emphasize that this interaction represented an
emerging form of languaging for both Mary Ann and her students. In
other words, dialogic interaction was not yet a classroom norm for
talking about mathematics for neither teacher nor student.
In concert with Peressini and Knuth (in press), I wish to clarify
my position that univocal discourse does have its place in the
classroom, albeit not at the expense of dialogic discourse. Their
conclusion that “all dialogic text must contain some univocal
functioning in order for clear communication to take place”
underscores the functional dualism of text argued by Lotman (1988).
However, as evidenced by Mary Ann’s early practice, there is a need
to cultivate balance in the function of text so that dialogic
interactions constitute a meaningful part of classroom discourse. The
discourse that characterized much of the problem-solving day
seemed to edge toward that preferred balance in which univocal and
dialogic discourse dualistically exist.
The routines enacted by Mary Ann and her students once
Problem 1 had been posed differed from those observed in her early
practice. Rather than giving hints and questioning incrementally to
lead students to a correct solution, Mary Ann enacted a “solicitation
routine”. In other words, she initiated the discussion by soliciting
solutions and procedures from students, focusing on their ideas and
strategies rather than her own. Furthermore, she seemed more
inclined to address students’ inappropriate responses by questioning
rather than telling (e. g., 78-85). In the absence of Mary Ann’s
routines such as giving hints, students were no longer obligated to
try to guess her thinking. Instead, they could share their solutions
and strategies as she requested. The pattern of interaction
constituted by these routines reflected a more interactive form of
languaging than that expressed in previous classroom discourse.
While this pattern was not fully adopted on the problem-solving day,
it did signal a shift in Mary Ann’s practice from the manner in which
student- or teacher-posed problems had typically been discussed.
This pattern is summarized below:
Emerging Pattern of Interaction
teacher writes the problem on the OP and asks students to
work in dyads for a solution
teacher asks various dyads for their solutions
student representative of each dyad responds
teacher selects dyad to explain their strategy
dyad representative responds
teacher comments, then selects another dyad to explain their
strategy
dyad representative responds
teacher comments, then selects another dyad to explain their
strategy
dyad representative responds
teacher questions dyad in order to understand their process
dyad representative responds
teacher selects another dyad to explain their strategy
dyad representative responds
teacher questions dyad in order to understand their process
dyad representative responds/clarifies thinking
teacher selects another dyad to explain their strategy
dyad representative responds
teacher questions dyad in order to understand their process
dyad representative responds/clarifies thinking
teacher addresses the validity of the various approaches.
During the remainder of the lesson, Mary Ann enacted the
familiar incremental questioning routine of asking leading, closed
questions (e. g., [What is] the inverse of divide?, Eleven minus five
is...?) to demonstrate similar problems to the whole class. Even so,
the experience of students being more actively engaged in discourse
seemed to open Mary Ann’s thinking to the value of dialogic
interactions. As she later reflected on the events that transpired
during this lesson, she wrote,
[At the beginning of the lesson], instead of throwing
information out, I let them figure the problem out in their own
style....To my surprise, one of the students performed the
problem exactly as the strategy suggested. Boy, was this a
memorable event. The pressure was lifted off of me.... Once
the students saw how one of their peers was able to solve the
problem, things were a lot more clear to all. I learned that
having the student come up with the solution means more to
the others than the teacher giving a long, drawn-out lecture.
This reflection supports the assertion that Mary Ann’s
pedagogical content knowledge was mediated toward a more
student-centered practice in the intermental context of the
classroom. In particular, where once she felt the obligation to give a
“long, drawn-out lecture” by “throwing information out”, she now
seemed to appreciate students thinking through a process with their
peers without a barrage of instructional questions from the teacher.
Moving forward in classroom discourse: Learning to listen.
Although Mary Ann’s emerging pedagogical content knowledge
exhibited a nonlinearity as she shifted between familiar and
unfamiliar routines, the events of the problem-solving day seemed to
anchor her flexibility for risk-taking in future discourse. An episode
several weeks later underscored this continuing growth in the
pattern and function of discourse in Mary Ann’s classroom. In an
investigation of the number of diagonals in a polygon, pairs of
students were given geoboards on which they were to form a
polygon (and all of its diagonals) by attaching rubber bands. As
students worked, Mary Ann recorded their findings on the board in
two columns, one showing the number of sides for a given polygon,
and the other its corresponding number of diagonals. After
determining the number of diagonals in a triangle, quadrilateral,
pentagon, and hexagon, students were asked to find a pattern that
would predict the number of diagonals in a heptagon without using
the geoboards. The following episode chronicles their ensuing
discussion.
97 Teacher: I want you to come up with a prediction, or a way
that we can figure out how many diagonals a heptagon
has without actually doing it on a geoboard. (Students
begin raising their hands.) I want everybody to
have a chance to think. Put your hands down.
Everybody talk with your neighbor. Think of a way....
I don’t want anybody forming a heptagon on the
board. I want you to do it thinking. Use your brain. (A
student asks a question that is inaudible to me.) No, just
talk it over with your partner. Y’all always want
to talk. I’m giving you a chance to talk.
The immediacy with which students appealed to Mary Ann for
some type of feedback and her consequent obligation to explain her
expectation that students negotiate with their partners suggests that
peer mediation did not yet constitute a shared understanding
between teacher and students for doing mathematics. Nevertheless,
Mary Ann persevered. After a reasonable amount of time had
passed, she congregated students’ attention for a discussion of their
thinking.
98 Teacher: So does everybody have a prediction, or has
formed a hypothesis, maybe?
99 Students: Yeah.
100 Teacher: Did you test it to see if it works?
101 Student 5: Yeah, it works.
102 Teacher: O. K., Susan what was your prediction? What do
you think about how many diagonals it’s going to have?
103 Susan: Fourteen.
104 Teacher: Fourteen. O. K., so Susan’s prediction is fourteen.
O. K., somebody else. Karen?
105 Karen: Fourteen.
106 Teacher: O. K., Karen thinks it’s going to be fourteen. O. K.,
Randy?
107 Randy: Fourteen.
108 Teacher: Fourteen. Christie?
109 Christie: Fourteen.
110 Teacher: Fourteen.
111 Student 6: People are raising their hand with the same
answer.
112 Teacher: Well, that’s fine. I want to hear what everybody
says.
113 John: I think twelve.
114 Teacher: You think twelve? Do you have something? Do you
have something to back up your prediction? [Some
way] how you want to test your hypothesis?
115 John: I did, but it’s wrong.
116 Teacher: Well, maybe not. Maybe if we test it. Lisa?
117 Lisa: I have twelve.
118 Teacher: O. K., so Lisa and John think it’s twelve. (Mary Ann
writes “12” on the board.) O. K., so John and Lisa,
since y’all got twelve, tell me how you got twelve.
119 John: Well, each one of those says increased by, uhm, a
number higher than each other, but...
120 Teacher: O. K., I didn’t understand that. So, you must have
said that not in my lingo. So, break it down.
121 John: Each one, when it’s increased by two, then
increased by another, each one is increased by one
and (at this point, John’s words become inaudible to
me) the number increased.
122 Teacher: Whoa! O. K., so you go....
123 John: You increase by two, then you add another one and
increase again by two. It’ll increase by three.
124 Teacher: O. K., so tell me. This (indicating one of the values
in the column containing the number of diagonals)
will increase by two. So we increase by two here.
Now tell me where I go from there.
125 John: Then you add one and increase by two again. Then
it increases by three.
126 Teacher: O. K., so you’re saying this increased by three. So
you add one to the former?
127 John: Yeah, and you keep doing that.
128 Teacher: Increase by four. So then what would your number
be here (indicating the unknown number of
diagonals)? If your prediction is you add one for every
time you add one here?
129 John: Fourteen. (His previous solution was twelve.)
130 Teacher: Fourteen. O. K., so somebody tell me, is there a
way, so you’re saying we’re going to have five here,
right? So how could we set this up in equation form to
get this number here (the unknown)? Is there a way
that you figured that out? (Susan raises here
hand.) Susan? (Her response is inaudible to me.) So is
this going to keep going in this order (indicating the
difference in consecutive numbers of
diagonals)? Two, three, four, five, six? Jack? (Jack
does not respond and Susan raises her hand again.)
Uhm, no, Jack is supposed to answer this. (After no
response from Jack, she turns to the class.) So we’re
saying that a heptagon is going to have fourteen
diagonals. O. K., if your hypothesis is correct, you’ve got
to back it up mathematically. So you’ve got to show me
an equation that can back this up, saying it’s
fourteen. If I’m solving for a variable, if this was
the unknown (she indicates the number of diagonals
of a heptagon).... O. K., so get out a pencil and
piece of paper and start computing. You’re
mathematicians and you’re scientists, and if
somebody asks you to test your hypothesis and
formulate your hypothesis, you’ve got to have some way
to back it up. I didn’t say this is right (i. e., fourteen).
I said this is what you’re making me buy into, or selling
to me. (After students start to work, she turns back to
John.) Do you have an answer?
131 John: Well, I have a hypothesis.
132 Teacher: O. K. (John’s response is inaudible to me.) O. K.,
John figured out that it was...fourteen.
133 John: Because it starts out at zero. Then you add one.
Then you add three on. Then you add four on.
134 Teacher: When you add nine and five, what do you get?
135 Student 7: Fourteen.
136 Teacher: Fourteen.
137 John: Oh, I thought it was twelve.
138 Teacher: (To Student 7) O. K., you say it’s fourteen. Prove it
to me.
For the remainder of the lesson, Mary Ann and her students
continued with this rich pattern of interaction. It stands in marked
contrast to the discourse that characterized her early practice. In
this episode, we see Mary Ann’s early tendency to ask leading
questions in order to demonstrate her thinking replaced with a
purpose to ask questions that make sense of students’ thinking. She
seemed to be learning to listen to her students dialogically. That is,
she seemed to be listening in order to generate new understanding,
not just determine if information had been correctly transmitted and
received (e. g., 120-128). This offers a compelling argument for Mary
Ann’s development as a teacher. Moreover, such discourse required
her to cede authority to her students, as she did with John. While she
risked vulnerability in doing this, her effort illustrates an ongoing
attempt to promote meaningful discourse in her practice.
As on the problem-solving day, Mary Ann again initiated a
routine of soliciting students’ solutions in the whole-class discussion
surrounding the problem of finding the pattern. Furthermore, where
earlier she may have solicited only correct solutions, it was the
introduction of an incorrect answer in this episode that finally got
her attention (102-114). This is not to say that correct thinking is not
a valued part of discourse. Indeed it is, and to suggest otherwise is
somewhat misleading. However, the activity of teaching must
extend beyond demonstrating correct procedures to include dialogic
interactions as well. Mary Ann’s later practice seemed to recognize
this need.
Mary Ann’s Students: More Knowing Others?
As a prospective teacher, Mary Ann was acculturated into a
mathematical community in which her students were already
members. Thus, students’ cognizance of that community’s existing
norms for doing mathematics positioned them as her more knowing
others. Clearly, Mary Ann’s students did not hold an overt agenda for
shaping her practice. Nonetheless, her sensitivity to students’
experiences while under her tutelage did yield a form of influence to
them. In particular, the early patterns of interaction observed in
classroom discourse led to a “reciprocal affirmation” of the
respective roles of teacher and student in the classroom. That is, the
cognitively simple questions that Mary Ann asked as she funneled
students toward a correct solution were often easily answered by
students (e. g., 11-17). As a result, students were affirmed in their
ability to do mathematics and their responses seemed to affirm Mary
Ann’s early practice.
The intermental context of the classroom thus served to direct
Mary Ann’s early languaging toward a more traditional paradigm of
giving information and inspecting the accuracy of transmission. In
effect, it mediated her intramental thinking about teaching
mathematics, that is, her pedagogical content knowledge. Indeed,
Vygotsky’s (1986) assertion that “higher voluntary forms of human
behavior have their roots...in the individual’s participation in social
behaviors that are mediated by speech” (p. 33) rang true for Mary
Ann’s early practice.
Interrupting the inertia that developed in univocal interactions
between Mary Ann and her students to make room for dialogic
discourse seemed crucial for her development. However, this
required Mary Ann to renegotiate classroom norms for doing
mathematics, to move away from rote question-and-answer
exchanges and toward interactions that probed students’ thinking.
Naturally, this met with initial resistance from students because they
were expected to assume an unfamiliar role in doing mathematics (e.
g., 55-58, 97). The resulting tension seemed to present a pivotal
juncture in Mary Ann’s development. It suggests a crucial point at
which other mediating agencies (e. g., university supervisor) can use
instructional assistance to support a prospective teacher’s efforts to
change what it means to do mathematics in his or her classroom.
Discussion
In its defense of new perspectives on teaching, the NCTM
Professional Standards for Teaching Mathematics (1990) outlined a
number of changes in teachers’ thinking needed to foster students’
intellectual autonomy. The shifts championed by this document
include a move toward verification through logic and mathematical
evidence and away from the teacher as the mathematical authority,
toward mathematical reasoning and away from memorization, and
toward hypothesizing and problem-solving and away from rote
answer-finding. Such recommendations must seem daunting to the
prospective teacher rooted intramentally in traditional norms of
doing mathematics. Indeed, change is not an easy process.
However, discourse in Mary Ann’s classroom did document an
emerging practice consistent with the views sanctioned by this
NCTM document. In particular, the univocal discourse that
characterized early languaging in her classroom was later tempered
with Mary Ann’s efforts to interact dialogically as she encouraged
students to hypothesize (e. g., 98-100) and justify their thinking with
mathematical evidence (e. g., 114-129) in order to solve non-routine
problems (e. g., 97). The patterns in classroom discourse expressed
this transition in Mary Ann’s pedagogical content knowledge as well.
Her image of the teacher as a mathematical authority, obligated to
funnel students exclusively to her own interpretation of a problem
through such routines as giving hints and incremental questioning,
gave way to a perception of the teacher as an arbiter of students’
ideas, obligated to solicit students’ thinking as a platform for
resolving mathematical dilemmas.
It is not my intention here to attribute such changes in Mary
Ann’s practice exclusively to classroom interactions. Rather, it is to
document the nature of such interactions and how they mediated her
pedagogical content knowlege. Contexts external to the classroom
shaped her practice as well. This raises an important issue that I
interject here and will pursue in Part IV. That is, how can teacher
educators provide the necessary scaffolding for the prospective
teacher so that mediation in the context of classroom discourse can
lead to a more effective practice?
The pattern and function of mathematical discourse in Mary
Ann’s classroom indicated her construction of pedagogical content
knowledge during the professional semester. In essence, Mary Ann’s
emergent languaging gave voice to the development of her
intramental thinking about teaching mathematics. Furthermore,
language in the intermental setting of the classroom mediated her
thinking about teaching mathematics because it exposed the nature
of students’ experiences in both affect and content. Therefore, the
dialectical role of language as articulated by Holzman (1996) was
evidenced in Mary Ann’s developing practice.
Implications of this study for teacher education center on
discourse. Specifically, we need to help prospective teachers
cultivate a practice that engages students in dialogical, as well as
univocal, classroom interactions. For the prospective teacher,
changing the nature of classroom interactions requires confronting
existing norms for doing mathematics. The resulting conflict places
students in a position to mediate the prospective teacher’s practice.
This is a critical juncture at which teacher educators can assist
prospective teachers in renegotiating the nature of classroom
discourse.
Furthermore, while the professional semester is an optimal
context to provide such assistance, the nature of discourse in a
prospective teacher’s classroom should be addressed in earlier
undergraduate settings as well. Indeed, the tool of language merits
the same attention in teacher education that physical tools (e. g.,
manipulatives) often enjoy. Ultimately, the mathematics teacher’s
ability to open a student’s zone of proximal development rests on the
nature of classroom discourse.
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APPENDIX
COOPERATING TEACHER ASSESSMENTOF THE
STUDENT TEACHER/COOPERATING TEACHER PARTNERSHIP
What were your goals and expectations when you entered this partnership?
How have these goals and expectations changed, if at all, during this practicum?
How did you perceive your role as cooperating teacher when you entered this partnership?
How has this perception changed, if at all, during this practicum?
Describe the nature of your partnership.
What do you think your student teacher learned from you?
Was there evidence that he or she successfully completed your perception of the practicum? If so, what?
What did you learn from your student teacher?
Describe your interactions with your student teacher. (E.g., Did you meet on a regular basis? Informally or formally? How did you negotiate your respective roles in the class?)
PRELIMINARY CODING SCHEME FOR DISCOURSE ANALYSIS
This scheme was developed based on the purpose of the
teacher’s utterance as ascertained during her conversational turns
in classroom discourse about mathematics. Multiple codes were
sometimes assigned to each utterance.
DQ: (Direct Question) Teacher asks a question to a particular
student.
RFI: (Request for Information) Teacher asks student(s) to
provide information that requires only rote recall (e. g.,
give definitions, acknowledge teacher’s solutions, respond to closed
questions).
RFPA (Request for Peer Assistance) Teacher asks other
student(s) to answer a question that a particular student
cannot answer.
RFC: (Request for Computation) Teacher asks student(s) to
perform a simple computation.
RFP: (Request for Procedure) Teacher asks student(s) to
explain procedure for obtaining a particular solution.
Not the same as RFJ (Request for Justification).
TCSR (Teacher Clarifying Student’s Response) Teacher poses a
question that paraphrases or repeats a student’s
response in order to verify her (teacher’s) understanding.
TP (Telling Procedure) Teacher tells/states a procedure or set of
fact(s) as a way of explanation, giving information, or
clarifying.
DRFPA(Denied Request for Peer Assistance) Teacher focuses on
one student’s participation when other peers are offering to
assist.
CFQ (Check for Questions) Teacher asks if student(s) has any
questions about a particular problem, or in general.
QSS (Questions Suggests Solution) Teacher asks leading
questions.
DP(Describing a Problem) Teacher is describing a problem for the
class to solve (e. g., reading a problem from the text) or
explaining directions for an activity.
CE(Communicating Expectations) Teacher is explaining what she
expects students to do in terms of homework, class
participation, and so forth.
HOA (Hands-on-Activity) Teacher uses a hands-on-activity.
This is to give additional information about a problem
students may be solving.
RFPS (Request for Problem Solving) Teacher asks student(s) to
solve a problem that requires higher order thinking
(beyond simple computation). May involve, e. g., modeling a
process with an equation, in order to solve.
RFJ (Request for Justification) Teacher asks student to justify
his or her thinking.
RTR (Request to Replicate) Teacher asks student(s) to
replicate a procedure with at most a minor alteration.
RTI (Request to Interpret) Teachers asks student(s) to
interpret information in order to answer a question.
THE CYCLE OF MEDIATION: A TEACHER EDUCATOR’S
EMERGING PEDAGOGY
Maria L. Blanton
North Carolina State University
Abstract
This investigation explores the pedagogy of educative
supervision in a case study of one prospective middle school
mathematics teacher during the professional semester. Educative
supervision as defined here uses the context of the prospective
teacher’s practice to challenge his or her existing models of
teaching. It rests on the Vygotskian (1978) tenet that the university
supervisor can guide the prospective teacher’s development to a
greater extent than the prospective teacher can when working alone.
Classroom observations by the university supervisor, teaching
episode interviews between the supervisor and prospective teacher,
and focused journal reflections by the prospective teacher were
coordinated in a process of supervision postulated here as the cycle
of mediation. The pedagogy of the teaching episodes, a central part
of this study, was closely aligned with instructional conversation
(Gallimore & Goldenberg, 1992).
The cycle of mediation suggests an avenue for effecting
prospective teachers’ development in the context of their practice. In
this study, perturbations experienced by the prospective teacher in
classroom discourse presented opportunities in supervision to
promote change in her practice. Moreover, instructional
conversation in the teaching episodes seemed to open the
prospective teacher’s zone of proximal development so that her
understanding of teaching mathematics could be mediated with the
assistance of a more knowing other.
Introduction
No one would seriously question the complexities of the
student teaching practicum. From a sociocultural perspective, the
practicum reflects the integration of often dissonant agendas of
teaching and learning that ultimately define a community into which
the student teacher is acculturated. It demands that the prospective
teacher negotiate tensions imposed by the juxtaposition of school
and university cultures in the context of a practice still in its infancy.
It is from the surfeit of pedagogical beliefs and practices constituting
this community that the student teacher’s practice emerges.
Despite these challenges, the practicum still promises the
optimal setting in which knowledge of content and pedagogy
coalesce in the making of a teacher. This possibility invites questions
about the ability of any agencies associated with the practicum to
effect teacher change. Of particular interest here is the role of
university supervision in that process. Specifically, is supervision an
effectual path to teacher development?
Furthermore, does supervision function as teacher education, or
does it instead reinforce pre-existing habits of teaching by focusing
on ancillary issues? Research on the supervision of student teachers
has produced a continuum of responses to these questions. While the
more skeptical suggest that we abandon supervision altogether
(Bowman, 1979), others argue that we must fundamentally alter the
way we supervise if we are to effect real change in the ways that
student teachers teach (Ben-Peretz & Rumney, 1991; Borko &
Mayfield, 1995; Feiman-Nemser & Buchmann, 1987; Frykholm,
1996; Richardson-Koehler, 1988; Zimpher, deVoss, & Nott, 1980).
Rethinking the Role of Supervision:
Education or Evaluation?
Historically, the role of supervision has likely tended toward
evaluative rather than educative interactions with student teachers.
That is, the traditions of supervision may be more closely described
by a perfunctory assessment of existing habits of teaching, buried
within an attention to classroom bureaucracy, rather than prolonged
interactions purposed to challenge those existing habits. Quite
possibly, this emphasis is a reflection of the chronological placement
of student teaching at the end of academic teacher preparation.
Furthermore, case loads that leave little time for one-on-one
interaction between the supervisor and student teacher often
relegate the supervisor to an evaluative role. However, Feiman-
Nemser and Buchmann (1987) challenge us to reconceptualize the
practicum (and hence supervision) as preparatory to future learning,
as educative rather than evaluative.
Research indicates that an educative stance is not currently
assumed in all supervisory relationships. In an investigation of
guided practice interactions between university faculty, cooperating
teachers, and student teachers, Ben-Peretz and Rumney (1991)
pinpointed the lack of professional reflection provided by support
personnel. They found instead that the authoritative demeanor
adopted by supervisors was met with passivity from student
teachers, resulting in little change in practice.
Elsewhere, Borko and Mayfield (1995) found that supervisors
focused on superficial aspects of teaching (e. g., paperwork, lesson
plans, behavioral objectives) and avoided in-depth discussions about
content and pedagogy, offering student teachers no specific
directives on how to change their practice. Concluding that
supervision seemed to exert little influence on student teachers’
development, they proposed instead that supervisors should actively
participate in student teaching and “challenge student teachers’
existing beliefs and practices and model pedagogical thinking and
actions” (p. 52). These recommendations might be seen to conflict
with the obvious physical parameters that constrain supervision.
However, an evaluative approach does not seem to engender
substantive change in teaching. In short, actively participating in
student teaching requires more than peripheral commitments by the
supervisor, but the result can be a practicum that functions as
teacher education rather than teacher evaluation.
Why should we consider an approach to supervision that
challenges student teachers’ models of teaching in the context of
their practice? First, it is within the demands of the classroom that a
student teacher’s internalized models of teaching are most readily
revealed (Feiman-Nemser, 1983). Such models, acquired through
years of classroom observations as a student, will persist throughout
the practicum if left unchallenged. Furthermore, any assumption
that desirable teaching habits necessarily derive from practice is
directly contradicted by existing research. For instance, Feiman-
Nemser cites studies in which successful student teaching was most
often equated with achieving utilitarian goals affiliated with
classroom management. This perspective on successful teaching
could likely impede any designs by teacher education programs to
infuse theory into practice. Feiman-Nemser and Feiman-Nemser and
Buchmann (1987) also report that student teachers tend to imitate
the persona of the school community into which they are
acculturated. Such behavior might reflect the specific habits of the
cooperating teacher or the more general attributes of the school
bureaucracy. Whether good or bad, this tendency could persist in the
absence of supervision that challenges student teachers’ models of
teaching.
Taken together, these findings point to supervision as pivotal
to teacher change. It is the supervisor who is most able to “provide
support and guidance for student teachers to integrate theoretical
and research-based ideas from their university courses into their
teaching” (Borko & Mayfield, 1995, p. 517). However, meaningful
supervision rests on reinterpreting the role of supervisor as teacher.
Sporadic visits by a supervisor whose primary function is to evaluate
peripheral characteristics of teaching seems to be an ineffective
route to changing practice (Borko & Mayfield, 1995; Frykholm,
1996; Zeichner, 1993).
From this position, I consider the place of educative
supervision in changing one student teacher’s practice. Educative
supervision is broadly defined here to mean supervision that
prioritizes the development of a student teacher’s pedagogical
content knowledge. Although student teaching is one of the most
widely studied components of formal teacher preparation, the
influence of supervision (particularly educative supervision) on
teacher learning is still unclear (Borko & Mayfield, 1995). Moreover,
understanding what educative supervision might resemble from a
sociocultural perspective remains virtually unexplored. As such, the
present study is guided by the following research questions:
1. What emerges as the pedagogy of educative supervision
during one prospective teacher’s professional semester?
2. What are the indications of the student teacher’s
development within the zone of proximal development?
Designing An Educative Approach to Supervision
As conceptualized in this study, educative supervision rests on
the Vygotskian (1978) tenet that the supervisor, as a more knowing
other, can guide the student teacher’s development to a greater
extent than the student teacher can when working alone. This
notion, theorized by Vygotsky as the zone of proximal development,
is unique in that it “connects a general psychological perspective on
[the individual’s]...development with a pedagogical perspective on
instruction” (Hedegaard, 1996, p. 171). As such, it lends theoretical
support to the use of intentional instruction during supervision.
Collecting the Data: The Cycle of Mediation
In order to study the influence of educative supervision on the
student teacher’s construction of pedagogical content knowledge, I
became the university supervisor of Mary Ann, a prospective middle
school science and mathematics teacher. Mary Ann was in her final
year of a four-year teacher education program for which ongoing
reforms in mathematics education are the unofficial mantra. She had
successfully completed her academic program and was eager to
begin the professional semester. Assigned to a seventh-grade
mathematics classroom in an urban middle school, Mary Ann was
paired with a veteran cooperating teacher who proved to be
extremely supportive. The cooperating teacher’s approach of sharing
her own wisdom of practice without stifling Mary Ann’s ideas led to a
positive, open relationship between them.
Mary Ann and I arranged weekly visits during the practicum
for what I conceptualized as an extension of Steffe’s (1991)
constructivist teaching experiment. That is, rather than eliciting
models of children’s constructions of mathematical knowledge, I was
interested in a teacher’s (i. e., Mary Ann’s) construction of
pedagogical content knowledge. Each visit consisted of a three-hour
sequence that began with an observation of Mary Ann teaching her
first period general mathematics class. Field notes taken during this
observation focused on episodes of discourse that reflected the
nature of her thinking about teaching mathematics. Immediately
following the observation, I collaborated with Mary Ann in a 45-
minute teaching episode to help make sense of these classroom
interactions. In particular, Mary Ann’s thinking about the
interactions, what they suggested about how students learn
mathematics, and consequently how subsequent lessons might be
modified to reflect this, were discussed. The visit concluded with a
second classroom observation of Mary Ann’s third period general
mathematics class. This provided the chance to document short-term
changes in Mary Ann’s practice as she taught the same subject to a
different class after a teaching episode. Additionally, Mary Ann was
asked to keep a personal journal in which she reflected on what she
had learned about her students, about mathematics, and about
teaching mathematics (see Appendix). Other written artifacts, such
as lesson plans, activity sheets, and quizzes, were collected as well.
At the conclusion of each visit, I audiotaped personal
reflections about emerging pedagogical content issues and how
future visits could incorporate these themes as learning
opportunities for Mary Ann. In all, I had eight visits followed by a
separate exit interview. Finally, I conducted two clinical interviews
with the cooperating teacher to obtain a more complete picture of
Mary Ann’s social context. These interviews were based on questions
concerning the cooperating teacher-student teacher partnership
that the cooperating teacher was asked to reflect on prior to the
meetings (see Appendix). Each visit, documented through field notes
and complete audio and audiovisual recordings, along with
supporting written artifacts and interviews with the cooperating
teacher, provided the data corpus for this investigation. The
supervisory process of observation, teaching episode, observation,
and written reflection that Mary Ann experienced as part of this
study is described here as the cycle of mediation (see Figure 1). It is
postulated in this study as a model for educative supervision.
Figure 1. The cycle of mediation in an emerging practice of teaching.
Pedagogy of the Teaching Episodes
The teaching episodes with Mary Ann were central to the
supervisory process outlined above. According to Steffe (1983), the
teacher’s role in an episode is to challenge the model of the
student’s knowledge and examine how that model changes through
purposeful intervention. This is consistent with the Vygotskian notion
of opening a student’s zone of proximal development and effecting
conceptual change through instructional assistance by a more
knowing other. Moreover, Manning and Payne (1993) suggest that
“in the teacher education context, this more experienced person is
likely to be a supervising teacher, college supervisor, teacher
educator, or a peer who is at a more advanced level in the teacher
education program” (as cited in Jones, Rua, & Carter, 1997, p. 6).
Intellectual honesty further mandated the pedagogy of the
teaching episodes. That is, since my purpose was to teach Mary Ann,
my own practice needed to be consonant with current reforms in
mathematics education. However, little is known about what it
means to supervise from this theoretical orientation. Moving away
from an authoritative voice, I turned to instructional conversation as
the underlying pedagogy. Instructional conversation stems from a
cultural ethos that emphasizes the use of narrative in an individual’s
development. Gallimore and Goldenberg (1992) and Rogoff (1990)
describe it as a primary means of assisted performance in preschool
discourse between parent and child. One’s way of life, embedded in
picture books and bedtime stories, is taught through conversation in
the context of familial relationships.
While formal schooling may seem far removed from this
setting, the essence of instructional conversation is a promising
technique in that context as well. Gallimore and Goldenberg (1992)
recognize that, traditionally, this form of teaching abates in school,
where teachers are more likely to dominate interactions and
students are less likely to converse with their teacher or peers. Part
of the difficulty of instructional conversation in the classroom is that
it involves the “paradox of planful intention and responsive
spontaneity” (Gallimore & Goldenberg, 1992, p. 209). Furthermore,
it requires teachers to shift from an evaluative role grounded in
“known-answer” questioning, to a facilitory role in which they elicit
students’ ideas and interpretations. Despite these challenges,
instructional conversation seemed an appropriate pedagogy with
which to engage Mary Ann in developing her craft.
Data Analysis
The teaching episodes with Mary Ann were selected as the
primary data source in this portion of the study. In order to
instantiate the pedagogy of these episodes as instructional
conversation, complete transcripts of four of the episodes were
coded by conversational subject using each speaker’s turn as the
basic unit of analysis. (See Appendix for a complete description of
the coding scheme.) The episodes were then quantified by a word
count to determine the emphasis given to each subject code and to
establish the amount of conversational time used by the university
supervisor (myself) and the student teacher. Additionally, transcripts
were examined for the use of known-answer questions and instances
of direct teaching by the supervisor.
Previously, I established evidence of long-term changes in
Mary Ann’s teaching during the practicum by documenting shifts in
the pattern and function of classroom discourse about mathematics.
Conclusions were based on the analysis of classroom observations
made during the practicum. In this portion of the study, the focus is
on the university supervisor as a mediating agent in the student
teacher’s practice. As such, transcripts from classroom observations
became a secondary source for corroborating short-term changes in
Mary Ann’s practice as a result of the teaching episodes. One visit,
selected as an exemplar of the cycle of mediation as a supervisory
model, was further analyzed to determine factors that promoted a
change in Mary Ann’s teaching. Specific excerpts from transcripts of
this visit (referred to later in the text as the “problem-solving day”)
are included to substantiate the results of the study.
Findings and Interpretations
Instructional Conversation in Teaching Episodes with Mary Ann
In an investigation of elementary students’ reading
comprehension, Gallimore and Goldenberg (1992) mutually
negotiated ten characteristics of instructional conversation: (a)
activating, using, or providing background knowledge and relevant
schemata; (b) thematic focus for the discussion; (c) direct teaching,
as necessary; (d) promoting more complex language and expression
by students; (e) promoting bases for statements or positions; (f)
minimizing known-answer questions in the course of the discussion;
(g) teacher responsivity to student contributions; (h) connected
discourse, with multiple and interactive turns on the same topic; (i) a
challenging but nonthreatening environment; and (j) general
participation, including self-elected turns. These characteristics
suggest what it might look like to supervise from a sociocultural
perspective. Those most representative of my instructional
conversations with Mary Ann motivate the following discussion on
how supervision from this perspective emerged during the
investigation. Excerpts from the problem-solving day are used to
situate these features within the context of the present study.
Activating, using, or providing background knowledge and
relevant schemata. Gallimore and Goldenberg (1992) maintain that
“students must be ‘drawn into’ conversations that create
opportunities for teachers to assist” (p. 209). An advantage of
educative supervision is that it can use the context of practice to
scaffold the student teacher’s emerging ideas about teaching. In
particular, episodes of classroom discourse became the nexus
between theory and practice in my instructional conversations with
Mary Ann. Using her classroom experiences as a referent seemed to
open her zone of proximal development and draw her into the
conversations. Indeed, Mary Ann became visibly passive when other
referents (e. g., my own experiences as a student teacher) were
introduced.
Thematic focus for the discussion. Gallimore and Goldenberg
(1992) also argue that “to open a zone of proximal development..., a
teacher has to intentionally plan and pursue an instructional as well
as a conversational purpose” (p. 209). By my third visit with Mary
Ann, I had identified a thematic focus on the nature of classroom
interactions that emerged after a mathematical task or question had
been posed. As discussed in Part III, observations prior to this visit
revealed predominantly univocal classroom interactions by which
Mary Ann funneled students toward her interpretation of the
problem at hand. The third visit presented an opportunity for
assisting Mary Ann in cultivating dialogic classroom interactions.
During the first period class that day, Mary Ann began a lesson
on “working backwards” as a problem-solving technique by giving
students a problem to work individually. “I’m thinking of a number”,
she said, “that if you divide by three and then add five, the result is
eleven.” After a short pause, Mary Ann began to dole out hints until
a correct solution appeared. After a student shared a procedure for
obtaining this solution, Mary Ann began a step-by-step account of
how to work backwards to find the answer. Analysis later showed
that she had interpreted students’ responses univocally, asking
cognitively-small questions (e. g., [What is] thirteen minus five?,
What is eight times three?) to align their thinking with her own.
Equating student feedback with understanding, Mary Ann’s
frustration surfaced later when the class attempted to solve a similar
problem.
1 Mary Ann: O. K., I’m thinking of a number [that], if you
divide by three and then add five, the result
is thirteen. So what would I first do just to get
an idea of what we’re talking about? Does
anybody know how we did the last one? (No one
responds to her questions.) O. K., what we need
to do first, step one, we need to write everything
down in the order in which we read it. So, we
start reading, “If you divide by three”, so we divide
by three. Then we’re going to add five. Then the
result is thirteen, and we want to work
backwards. So, what have we got to do when
we work backwards? (Again, students don’t
respond.) O. K., what was the word we used
when we talked about what we’ve got to do with all
of these [operations]?
2 Class: Inverse.
Univocal interactions between teacher and students continued
until a student produced a response of twenty-four. Mary Ann
concluded, “Twenty-four. So, that’s my answer. That is the answer. I
ask you what number did I start with, you’ll say what?” The students
were silent. She continued, “What number did I start with? The
problem says, ‘I’m thinking of a number’. What number am I thinking
of?” Hesitantly, students tried to guess the response, suggesting
various numbers that had occurred in the problem. Twenty-four
seemed to dominate, cueing Mary Ann to once again argue its
veracity. She repeated, “Twenty-four. That is your answer. You
worked backwards. You said thirteen minus five is eight and eight
times three is twenty-four.”
The perturbation that Mary Ann exhibited during this
interaction seemed to grow out of puzzlement that students did not
understand what she had carefully explained. This left her at a
pedagogical impasse. The challenge of the teaching episode that
followed (and future episodes) was to use such interactions to help
Mary Ann develop a sense of mathematics as a problem-solving
endeavor in which students struggled with unfamiliar problems and
justified their ideas through mathematical discourse with each other
and Mary Ann. In essence, the challenge was to help Mary Ann
create a classroom discourse in which dialogic and univocal
interactions dualistically existed. Using such interactions as a
thematic focus became an avenue for encouraging Mary Ann to
interact dialogically with her students. It provided an instructional
and conversational purpose that continued throughout the
practicum.
Direct teaching, as necessary. Given that students are more
likely to teach in ways they are taught (Borko & Mayfield, 1995;
Feiman-Nemser, 1983), I minimized instances of direct teaching in
the episodes with Mary Ann. Instead, I relied on “prompting,
modeling, explaining,...discussing ideas, [and] providing
encouragement” (Jones, et al., 1997, p. 4) to give structure to our
conversations. This emphasis is consistent with the pedagogy of
instructional conversation, which prioritizes students’ participation
in dialogue. Table 1 illustrates the amount of conversational time
used by Mary Ann during the teaching episodes. The results support
my intent to maintain a facilitory role that kept her at the center of
discourse. By this, it became Mary Ann’s responsibility to rethink her
teaching. What emerged was the opportunity for her to retain
ownership of her practice. Furthermore, this sense of ownership
seemed to heighten Mary Ann’s willingness to put new ideas into
practice.
Table 1
Conversational Time Used by Participants in the Teaching Episodes
Participant TE1 TE2 TE3 TE4
Student teacher 82 84 72 73
University supervisor 18 16 28 27
Note. Values represent percentage of time a given participant spoke
during a teaching episode. Percentages are based on word counts.
“TE” denotes a teaching episode.
Minimizing known-answer questions in the course of the
discussion. “When known-answer questions are asked, there is no
need to listen to a child or to discover what the child might be trying
to communicate” (Gallimore & Goldenberg, 1992, p. 209). An
imperative of the teaching episodes was to avoid the use of known-
answer questioning and instead, to interpret Mary Ann’s utterances
dialogically. As a result, the questions I posed to Mary Ann were
essentially open-ended. In the ensuing dialogue, Mary Ann was
expected to justify her thinking about teaching mathematics and her
consequent actions in the classroom, not passively respond to a
supervisor’s prompts.
Teacher responsivity to student contributions. The amount of
conversational time used by Mary Ann suggests that her
contributions were a priority in supervision (see Table 1). Moreover,
being responsive to her ideas required being sensitive to her zone of
proximal development as well. The following dialogue was
excerpted from an instance of intentional instruction with Mary Ann
during the problem-solving day. It illustrates the effort to maintain
sensitivity to her zone while guiding her thinking, to base
supervision on her understanding of teaching mathematics, not my
own.
3 Supervisor: Is this the kind of [math word] problem where you
could let two or three [students] work
together, and try to figure out how to do it, and
see what kind of method they come up with?
4 Mary Ann: That could be an idea. Maybe I could let them
work with the person beside them.
5 Supervisor:Do you think that is even feasible? If so, why? Or
why not?
6 Mary Ann: Two heads are always better than one, and the kid
next to you might be thinking of one way, but
might be stumped on how to do the next. But
you might be able to help him figure that out. The
only thing is that I don’t know if they (her
voice trails off). We’ll see, though. That might be a
way to try. I don’t know if they can handle that,
talking to each other.... They’re just talkers, all
the time. Maybe if I show them that they can
have some freedom like that (her voice trails off).
Mary Ann’s uncertainty toward this suggestion was manifested
as concern over classroom management. My role then became to
redirect the instructional conversation so that it was within her zone
of proximal development. This involved connecting her concerns
about students’ behavior with the alternative approach we were
negotiating (7).
7 Supervisor:Do you think they can handle working with a
problem that they can’t figure out, trying to solve a
problem in that sense?
8 Mary Ann: I think they would be more apt to keep their
attention on that problem if they’re working
with somebody rather than working by
themselves.
After probing Mary Ann’s understanding of the role of word
problems in mathematics and teaching mathematics, we revisited
the previous topic.
9 Supervisor:Would you be comfortable, for example, if you
came [in class],...[threw] out a problem, and [let]
students work it for a while, and try to figure
out how to come up with a solution?
10 Mary Ann: Yeah. That’s how I’m thinking about starting the
next class. We’ll have to go over homework
first because they’re having a quiz on that
tomorrow. And then just have that [math word]
problem up on the board, and then tell them
to solve it. Don’t introduce anything about working
backwards.
Transcripts strip the dimensionality of dialogue. Although it’s
not readily apparent, Mary Ann’s claim, “That’s how I’m thinking
about starting the next class” (10), was spoken with a sense of
reflection and ownership. It stood in sharp contrast with her initial
reticence (6). It should also be noted that this remark occurred over
halfway through the teaching episode, after much attention had been
given to Mary Ann’s thinking about problem solving and the nature
of interactions that surrounded a problem posed in class. While one
might argue that a didactical approach (in the American semantic) to
supervision would have been more efficient, I seriously question if it
would have led to Mary Ann’s commitment to try an alternative
strategy. However, instructional conversation seemed to open her
zone of proximal development cognitively and affectively, thereby
producing at least a short-term commitment to change.
Connected discourse, with multiple and interactive turns on
the same topic. Specific directives on how Mary Ann might alter her
instruction after a given mathematical task had been posed were
revisited several times within the teaching episode on the problem-
solving day. When I sensed that my directives were out of her zone
of proximal development, I steered to related subjects (e. g., her
perception of problem solving in mathematics), but eventually moved
back to this topic. Furthermore, this particular teaching episode
became a “hook” (Gallimore & Goldenberg, 1992), or referent, in
later conversations with Mary Ann.
Table 2 is provided as an overview of general topics addressed
in the teaching episodes. In particular, it summarizes the focus on
pedagogical content knowledge throughout Mary Ann’s practicum.
Specifically, instructional conversations with Mary Ann were
dominated throughout by discussions on topics coded as
“mathematics pedagogy”. This, coupled with discussions on other
subjects closely linked to pedagogical content knowledge (i. e.,
“mathematical knowledge” and “general pedagogy”), left little room
for the peripheral issues of school bureaucracy in our teaching
episodes.
Table 2
Conversational Time Given to Subject Code During Teaching
Episodes
Subject Code TE1 TE2 TE3 TE4
Mathematics pedagogy 20 51 53 47
General pedagogy 28 20 9
23
Mathematical knowledge 0 7 11
16
Knowledge of student 24 17 14 14
understanding
Classroom management 28 0 5
0
Student-teacher 8 5 8
0
relationship
Note. Values represent percentage of time the specified subject was
discussed in a teaching episode. Percentages are based on word
counts.
“TE” denotes a teaching episode.
A challenging but nonthreatening atmosphere. In action and
words, Mary Ann seemed at her ease during the observations and
teaching episodes. During a couple of observations, she even asked
for my input on a particular problem as she taught the lesson.
Moreover, the rapport we established early on seemed to contribute
to her responsiveness in the teaching episodes. Upon reflection, I
could have created a more challenging atmosphere for Mary Ann.
Indeed, the ongoing tension of supervision is understanding how to
strike an optimal balance that effectively challenges the student.
Instructional Conversation in Retrospect: More on the Problem-
Solving Day
The dissonance Mary Ann experienced in classroom
interactions presented opportunities in supervision to promote
change in her practice. However, she needed to own that change.
Instructional conversation in the teaching episodes became a conduit
to that ownership. On the problem-solving day, it seemed to extend
to Mary Ann a commitment to modify her practice. Mary Ann began
the lesson following the teaching episode as we had planned.
Departing from her previous strategy, she placed students in dyads
to solve the problem that had been assigned as individual seat work
in her earlier class. Removing herself as the sole authority, she
delayed closure so that students would begin to communicate
mathematically with each other. As one of the students began
explaining her group’s strategy for solving the problem, Mary Ann
looked at me in excited disbelief and mouthed, “Wow!” She
commended the student, “You just taught our lesson for today!”
Mary Ann’s expression told the story that her journal reflection later
confirmed.
Teaching this [to the first period class] was a real eye-opener
for me. I think I totally confused my students completely. I
tried to show them steps without letting them think about the
problem themselves.... [The next class] was different. After
[the university supervisor] and I talked about the lesson and
going over several suggestions, things seem [sic] to run much
smoother. Instead of throwing information out, I let them
figure the problem out in their own style.... To my surprise, one
of my students performed the problem exactly as the strategy
suggested. Boy, was this a memorable event. The pressure was
lifted off of me.... Once the students saw how one of their
peers was able to solve the problem, things were a lot more
clear to all. I learned that having a student come up with the
solution means more to the others than the teacher giving a
long, drawn-out lecture. Sometimes you need for things to flop,
so you can think up new ways to approach the situation.
From my observations, the problem-solving day was a first step
in Mary Ann’s attempts to interact dialogically with her students.
Furthermore, it seemed to anchor her willingness to take risks in her
practice based on ideas mediated through instructional conversation.
She continued to develop, albeit in a nonlinear fashion, toward a
practice which included dialogic as well as univocal interactions.
Discussion
This study investigates in part what it means to educate
student teachers from a sociocultural perspective during the
professional semester.
Cobb, Yackel, and Wood (1991) maintain that
teachers should be helped to develop their pedagogical
knowledge and beliefs in the context of their classroom
practice. It is as teachers interact with their students in
concrete situations that they encounter problems that call for
reflection and deliberation.... Discussions of these concrete
cases with an observer who suggests an alternative way to
frame the situation or simply calls into question some of the
teacher’s underlying assumptions can guide the teacher’s
learning (p. 90).
In this sense, the cycle of mediation became educative for
Mary Ann. Specifically, coordinating classroom interactions observed
during Mary Ann’s teaching with the instructional conversation of
the teaching episodes and Mary Ann’s reflections about her practice
converged to promote Mary Ann’s development within her zone.
Although such a process is arguably quixotic, it does suggest an
avenue for effecting prospective teachers’ development in the
context of their practice.
Furthermore, while it is difficult to establish a direct link
between instructional conversation and conceptual development
(Gallimore & Goldenberg, 1992), instructional conversation does
suggest an alternative pedagogy for educative supervision.
Specifically, it seemed to open Mary Ann’s zone of proximal
development so that her understanding of teaching mathematics
could be mediated with the assistance of a more knowing other.
Moreover, the notion that an individual’s intramental
functioning reflects the intermental context of the classroom
(Wertsch & Toma, 1995) suggests that instructional conversation
could mediate Mary Ann’s practice toward that type of pedagogy.
Simply put, students most likely teach in ways they are taught.
However, as a caveat, it should be noted that multiple influences
shape the prospective teacher’s emerging practice. This sometimes
limits, or even negates, the influence of the supervisor.
Understanding how all of these factors coalesce in the making of a
teacher is at best a delicate process. As such, this investigation is a
first attempt to understand that process from the supervisor’s lens.
References
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APPENDIX
EXPECTATION OF THE STUDENT TEACHER
Sept. 24
(Student Teacher)
As I have already mentioned to you, each of my observations will involve a (telephone) pre-conference, observing 2 classes weekly (when possible), and a post-conference. Except for the frequency, this should be typical for all student teachers. I would like for you to have a copy of your lesson plan to give to me on the day that I observe. Also, I would like a reflective journal entry for each of my visits. Below are some questions that I would like for you to address. Since this is only one entry per week (roughly), it should not be too demanding of your time. You do not have to do this separately from the journal requirements for Dr. S and/or Dr. N, but you may include my questions within their requirements (e.g., they may require more than one entry per week and you should fulfill that obligation, but I am only specifically looking for a single detailed entry corresponding to my visits that addresses the following questions. Where there is possible overlap, use it to your advantage.) If you have any questions, please let me know.
Questions to consider for your journal entries:
What student interaction(s) was/were the most memorable to you (during my observation)? (Please avoid interactions that deal with classroom management, etc. I am only interested in interactions as they relate to your teaching mathematics.) Why?
How (if at all) did this affect your instruction? How (if at all) did this affect your understanding of mathematics? What did you learn about (your) students as a result of this?
I have enclosed a consent form for you to sign. I will pick it up when I observe you.
Thank you!
Maria
COOPERATING TEACHER ASSESSMENTOF THE
STUDENT TEACHER/COOPERATING TEACHER PARTNERSHIP
What were your goals and expectations when you entered this partnership?
How have these goals and expectations changed, if at all, during this practicum?
How did you perceive your role as cooperating teacher when you entered this partnership?
How has this perception changed, if at all, during this practicum?
Describe the nature of your partnership.
What do you think your student teacher learned from you?
Was there evidence that he or she successfully completed your perception of the practicum? If so, what?
What did you learn from your student teacher?
Describe your interactions with your student teacher. (e.g., Did you meet on a regular basis? Informally or formally? How did you negotiate your respective roles in the class?)
CODING SCHEME FOR TEACHING EPISODES WITH MARY ANN
Mathematics pedagogy (MP). Conversation that addresses Mary
Ann’s learning about teaching mathematics as well as how she
teaches mathematics.
General pedagogy (GP). Conversation that addresses principles of
teaching that aren’t specific to mathematics (e. g., pacing
instruction, diversity in student learning).
Mathematical knowledge (MK). Conversation that addresses Mary
Ann’s knowledge about mathematics.
Knowledge of student understanding (KSU). Conversation that
addresses Mary Ann’s understanding of how students are or are not
understanding the content and how that directly affects her practice.
References to test performance are also designated KSU.
Classroom management (CM). Conversation that addresses non-
academic student needs (e. g., discipline, student health).
Student/teacher relationship (STR). Conversation that addresses
Mary Ann’s relationship with her students and how that influenced
instruction.
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APPENDIX
COOPERATING TEACHER ASSESSMENTOF THE
STUDENT TEACHER/COOPERATING TEACHER PARTNERSHIP
What were your goals and expectations when you entered this partnership?
How have these goals and expectations changed, if at all, during this practicum?
How did you perceive your role as cooperating teacher when you entered this partnership?
How has this perception changed, if at all, during this practicum?
Describe the nature of your partnership.
What do you think your student teacher learned from you?
Was there evidence that he or she successfully completed your perception of the practicum? If so, what?
What did you learn from your student teacher?
Describe your interactions with your student teacher. (E.g., Did you meet on a regular basis? Informally or formally? How did you negotiate your respective roles in the class?)
Consent to Release Information for Research Purposes
I, __________________________________________________, give permission for the contents of my journal, portfolio, surveys, videotapes, and any audiotapes to be used as a research resource for written professional reports. I also give my permission for information from interviews and conferences to be used. I understand that my name will never be used without my permission and that no information in written or verbal form can be used to punitively assess my student teaching performance. I also understand that all copies of audio tapes and video tapes will be destroyed two years from the end of the project. If any changes in this agreement are required, I must be contacted in writing.
Signature_____________________________________________Date_____________
Cooperating Teacher Agreement Form
I, ______________________________________________________, give permission for data collected from my classroom during this professional semester to be used as a resource for research of prospective education. I understand that the data collected will include videotaped classroom observations of the student teacher as well as interviews, surveys, and contents of the student teacher’s journal and portfolio. I also understand that my name will not be used in any way without my permission and that no identifying information in written or verbal form will be used. I also understand that copies of audiotapes and videotapes will be destroyed two years from the end of this project. If any changes in this agreement are required, I must be contacted in writing.
Signature___________________________________________Date________________
PRELIMINARY INTERVIEW PROTOCOL
Pick as many episodes from the classes I observed as I have time for. When possible, review these episodes on the video camera.
How did you feel about your lesson(s)?
What had you planned to do in this episode? Or had you planned anything?
What were you thinking during this episode?
Did you/would you change your instruction any as a result of this reflection? If so, how?
What did you learn about teaching in this episode?
What did you learn about mathematics?
What did you learn about teaching mathematics?
What did you learn about students? How will that affect your instruction?
EXPECTATION OF THE STUDENT TEACHER
Sept. 24
(Student Teacher)
As I have already mentioned to you, each of my observations will involve a (telephone) pre-conference, observing 2 classes weekly (when possible), and a post-conference. Except for the frequency, this should be typical for all student teachers. I would like for you to have a copy of your lesson plan to give to me on the day that I observe. Also, I would like a reflective journal entry for each of my visits. Below are some questions that I would like for you to address. Since this is only one entry per week (roughly), it should not be too demanding of your time. You do not have to do this separately from the journal requirements for Dr. S and/or Dr. N, but you may include my questions within their requirements (e.g., they may require more than one entry per week and you should fulfill that obligation, but I am only specifically looking for a single detailed entry corresponding to my visits that addresses the following questions. Where there is possible overlap, use it to your advantage.) If you have any questions, please let me know.
Questions to consider for your journal entries:
What student interaction(s) was/were the most memorable to you (during my observation)? (Please avoid interactions that deal with classroom management, etc. I am only interested in interactions as they relate to your teaching mathematics.) Why?
How (if at all) did this affect your instruction? How (if at all) did this affect your understanding of mathematics? What did you learn about (your) students as a result of this?
I have enclosed a consent form for you to sign. I will pick it up when I observe you.
Thank you!
Maria
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